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. is the
4 intersection of the family of prime ideals of I
S
containing
.
Proof. The proof of follows from
(2)
.
(1)
is elementary, and
We prove
suffices to show that
I
(2) .
By Zorn's Lemma, it
is prime if
respect to failure to meet
T .
(3)
I
is maximal with
Thus, assume that I u
{a} u
a,b
e S\I
.
Each of the
ideals
I u
{b} u
(b
+ S)
T , and hence there exist
s 2 ,s 2 e S
such that
quently,
a + s^ e T
a + s 1 + b + s 2 eT
follows that If ment
meets
S
.
b + s2 e T .
I
is prime.
is an additive monoid with identity (zero) ele s e
S , then
if there exists
x e
S
s is
such that
said to be invertible s
+ x = 0 ; the set
invertible elements of
S
forms a subgroup
Gis the largest subgroup of
S
containing
sum
S
°f elements of
each if
s^ isinvertible. G * S .
If
H
S\G
is any subgroup of H of
H .
lation
defined
by
a~
s + H
h 6H
S
is
, then
~
cancellative if a, be S .
The
be empty;
if
s
of a semigroup
s + a = s + b set
. A finite
S
containing
C
b
if
C
S
0 ,
a = b + h
s
S
S is said
implies
S
for and
e S .
a = b
to be for
of cancellative elements of
C * <J> , then
of
In fact, if the re
is an equivalence relation on
is the equivalence class of An element
and
induces a partition of
s + H
on
S,
is a prime ideal
into disjoint cosets ~
0
of
G
is invertible if and only if
Thus,
then as in the case of groups,
some
Conse
I n T = 0 , it
Since
a + b £ I , and hence
and if
of
0
and
(a + S)and
is a subsemigroup of
S
may
S
, a
5
a finite sum and hence
I1? ,s. is in i=l 1
S\C
C
if and only
is a prime ideal of
S = C , we say that
S
S
'
if
if each
s.e i
S * C .
C , *
If
is a cancellative semigroup.
A
familiar result from elementary group theory states that a finite cancellative semigroup is a group.
It is clear that
a subsemigroup of a group is cancellative.
Theorem 1.2
proves a form of the converse. THEOREM 1.2. semigroup
If C
S and if
is a subsemigroup of an
each element of C
S , then there exists an imbedding abelian monoid -f(c)
in
T
such that
T for each
(f(s) - f(c)|s e S
(1)
f
and
c e C} .
is cancellative in of
f(c)
c e C , and
additive
S
into an
has an inverse
(2)
T=
The monoid
T
is de
termined, up to semigroup isomorphism, by properties
(1)
and
T
(2) .
If
S
is cancellative and
S = C , then
is
a group. Outline of a pro of . the familiar one used tegers from the set from let
Zq
property of Denote by and let T
C
To wit,
let
A
of in
if Sl + c 2 = s 2 + c 1
and
defined by (thecancellative
theequivalence class of
(s,c)
is an abelian monoid with zero element
the mapping
Z
A - S * C
be the set of equivalence classes
the operation
is like
is used in proving transitivity of
[s,c] T
T
of nonnegative integers, starting
be the equivalence relation on
Csi,c i) ~ (s2,c2)
set
in constructing the ring
Peano’s postulates. ~
The construction of
~ ) under
[s,c]
.
* [c,c]
~
The under
[s^,c^] + [s2 ,c2] = [s^ + s 2 ,c^ + c 2 ] , and f : S -- ►
T
defined by
f(s) = [s + c,c]
is
6 an imbedding of [2c,c]
has
element
S
in
inverse
[s,c]
of
T .
If
c e C , then
[c,2c]
in
T
T
,and an arbitrary
can be written as
[s + c,c] + [c,2c] = f(s)
- f(c)
.
It is
determined up to semigroup isomorphism by and
(2) , and if
of
T
f(c) =
clear that T the properties
S = C , then an arbitrary element
has inverse
[c,s]
in
T
so that
is
T
(1)
[s,c]
is a group.
This completes the proof of Theorem 1.1. The monoid
T
constructed as in the proof of Theorem
1.2 is called the quotient monoid of
S
with respect to
By abuse of notation, we write the elements of form
s - c
instead of
be a subset of T
T .
If
f(s) - f(c) S
and
T
in the
we consider
S
to
is cancellative, then the group
of Theorem 1.2 is called the quotient group of
S ; to
within isomorphism, it is the smallest group in which can be imbedded.
C .
S
One way of classifying cancellative semi
groups is in terms of properties of their quotient groups. For example, abelian groups
arefrequently classified as
being either torsion— free, mixed, or torsion groups. recall the definitions. free if
0
An abelian group
is the only element of
G
is a torsion group if each element of and
G
tive semigroup with quotient group
of
is torsion-
of finite order; G
G
has finite order,
is mixed if it contains elements of infinite order
and nonzero elements of finite order.
that
G
We
G
If
S
is a cancella
G , then the condition
be torsion— free, for example, can be stated in terms
S ; in fact,
G
is torsion— free if and only if
satisfies the following condition.
S
7 (1.3)
For any positive integer
the equality
nx = ny
implies that
We note that the condition in
n
and any
x,y e S ,
x = y .
(1.3) makes sense whether
is assumed to be cancellative or not; finition of a torsion— free semigroup.
S
it is used as the de We will encounter
torsion— free semigroups in Section 3 in determining the class of (totally) ordered semigroups.
We avoid the use of the
term torsion semigroup, preferring the term periodic semi group instead.
The definition is given in Section 2 ; we
remark that for a cancellative semigroup is such that
S
S , the definition
is periodic if and only if its quotient
group is a torsion group. Section 1 Remarks The material in Section 1 is both basic and elementary, and hence complete proofs have not been given in the section. Some notation and terminology have been used without ex planation.
For example,
inclusion, and where
B
A\B
s
denotes inclusion,
denotes the complement of
is a subset of the set
group isomorphism,
< B
proper in
A ,
A ; the notions of semi
isomorphic semigroups,
and an imbedding of
one semigroup in another are defined in the expected ways. Many of the notions in Section 1 are meaningful, and have been studied,
for noncommutative semigroups.
But
Theorem 1.2 does not extend to the noncommutative case. Malcev in [94] constructed a noncommutative cancellative semigroup
S
that is not imbeddable in a group; he also
showed that the semigroup ring of
S
over the field
Q
of
rational numbers is a noncommutative ring without zero di-
8
visors that is not imbeddable in a field. Clifford and Preston’s book
[32] is a standard reference
on the theory of (not necessarily commutative)
semigroups.
In addition, Re d e i ’s book [122] treats finitely generated commutative semigroups, and especially the theory of con gruences on such semigroups. The
papers
[31],
[39] and [73]
are primarily concerned with the basic theory of commutative semigroups.
9 §2.
Cyclic Semigroups and Numerical Monoids
Two concrete classes of semigroups frequently encoun tered in the general theory are the semigroups
<s>
generated by a single element and submonoids of the monoid Zq
of nonnegative integers.
This section contains basic
results concerning semigroups in these two classes. An additive semigroup genic) if there exists {s,2s,...,ns,...} groups.
m
ms
and
s e
S such that
and
infinite order.
S = <s> =
are two cases to consider. ns
Z+
First,
may be distinct for distinct
n ; in this case
additive semigroup
mono
We determine the structure of such semi
Clearly there
the elements tegers
.
S is said to be cyclic (or
S
in
is isomorphic to the
of positive integers, and
The second case, where
S
ms = ns
has
for some
m * n , is covered in Theorem 2.1. THEOREM 2.1. group such that
Assume that ms = ns
for some
smallest integer such that let
S = <s>
is a cyclic semi
m * n .
ks = rs
Let
for some
k
be the
r < k , and
m = k - r . (1) For
and only
u > v > r ,the equality
ifm
divides
u
us = vs
- v .
(2)
S = {s,2s,...,(k - l)s}has cardinality
(3)
G ={rs,(r + l)s,...,(k
S ; its identity element is between
r
Proof. rs = rs + ms
and (1)
holds if
hs , where
k - l = r + m - l Assume that
- l)s}
k -
1 .
is a subgroup of h
is the integer
that is divisible by
u - v = ma .
, it follows easily that
m .
Since
rs = rs + mas . Hence
vs = rs + (v - r)s = rs + mas + (v - r)s = (v + ma)s = us .
10
For the converse, assume that {r,r + l,...,k - 1}
is a set of
there exist unique integers u
= u1
(modm)
and
proved shows that k
m
u^s
The choice of
k
this set so that .
The part of (1) already-
= us = vs = v^s
,
and the choice
v\
u
= v (modm)
*Therefore
also implies that
{s,2s,...,(k - l)s}
has cardinality
that it is equal to
S .
the mapping
Since
consecutive integers,
u i>v i
ve v^ (modm)
u\ =
implies that
us = vs .
Moreover,
as ----a + (m)
group isomorphism;--since
of
G
Z/(m)
into
(1) shows
also shows that Z/(m)
is a
is a group, then
also a group and its identity is
.
the set
k - 1 , and (1)
of
hs , where
semi
G
is
h + (m) =
0 + (m) . If
s,r
and
m
are as in Theorem 2.1, then s , and
refer to
as the index and period of the semi
group
and
m
<s> ; the integer
r + m -1
s , and also the order of the order of <s>
<s>
S
<s> . As
We also
is called the order of in the case of groups,
is the number of elements in
is finite, then
group If
<s>
is the period.
is
called the index of r
m
r
s
<s>
is said to be periodic, and
is periodic if each element of is infinite, then
s
S
If
a semi
is periodic.
is aperiodic, and
aperiodic if each non-identity element of
.
S
S
is
is aperiodic.
The index and period of a cyclic semigroup determine the cyclic semigroup up to isomorphism. tive integers C(r,m)
r
of index
and r
m and
Moreover,
, there exists
for any posi
a cyclic semigroup
period m .One such semigroup is
the multiplicative semigroup generated by the class of in
X
Z[X]/(Xr+m - Xr ) ; another is the multiplicative semi-
11
group in q
Z/(2m - 1)Z © Z/qrZ
is a prime greater than
is a groupiff r = 1 of course,
generated by
2m - 1 .
iff C(r,m)
C(r,m)
,
We note that
is amonoid;
the cyclic group of order
hand, each
(2,q)
m .
where
C(r,m)
C(l,m)
is,
On the other
contains a unique idempotent element
(in additive notation, an element
t
is idempotent if
t = t + t = 2t) . The additive semigroup
Z+
infinite cyclic semigroup and
of positive integers is the (Z^,+)
is the monoid ob
tained by adding an identity element to
Z+
as in Section 1,
On the other hand, the fundamental theorem of arithmetic shows that the multiplicative semigroup
Z+
is the internal
weak direct sum of countably many copies of
(Zq ,+) , in the
sense of the following definition. monoid with zero element submonoids of
0 .
S =
^eA^a^
^
sV ns i=l
equality n 7
E? ..s = E? nt 1=1 a1=1 a. 1 1
S
a
i .
, with
s
ai
for each
S .
finite, we write (Z + ,*) , let let
S^
e S
ai
S
°f
be the submonoid
we
is expressible
in
s
a. 1
i , and if an = t a. 1
S = E S a a
for
and that
a e A , but is not equivalent
S = S,©...©S . 1 n
p^ < p 2 (a - 1) (b - 1) , write
t > q^
be relatively prime posi
y
0 < i < a -1 .
a , the set If
and
n > (a - 1 ) (b - 1) , then there exist
nonnegative integers Proof.
a
r = rj * t^ien
ta + rj = n < a .
But
a contradiction. +qja
+ r. =jb.
+
Therefore
r^. = (t - q^)a + jb .
Theorem 2.2 implies that a numerical monoid containing two relatively prime integers contains an ideal
K + Zq =
13
{x e ZQ^X “Z 0
^0r some
K e Z+ .
The next result
shows that each nonzero primitive numerical monoid contains a pair of relatively prime integers. THEOREM 2.3. monoid, that
If
S
is a nonzero primitive numerical
S n Z+
There exists a finite subset
with
that
g.c.d. 1
.
1 = x ^ a ^ . . .+x a 11 nn
for
{a,,a0 .... a } l l n
Choose integers .
x^,...,xn
k sufficiently large, each
for each integer & x^ + kxn
and is relatively prime to r
THEOREM 2.4. let
d be the (1)
S
kd S
monoid,
Hence
(a.c.c.)
There exists a finite set
then
Proof.
C
' xn ^an
S . K
such that
S
k > K.
is finitely generated.
such that if
so
e S .
n
g.c.d.of the elements of
for each
k , and 7
be a nonzero numerical monoid and
ascending chain condition (3)
a
There exists a positive integer
contains (2)
Let
such
is positive
+ ^a )a^ = 1 + [k (a i + •••+an _
that
of
Then 1 + [k(a.. + ...+a ,) - x la 1 n -1 nJ n
(x, + ka )a, + ...+(x 1 + ka )a 1 n l n -1 n n -1
S
such
gcd{a,b} = 1 . Proof.
S
a,b e S
then there exist positive integers
ZQ
satisfies the
on submonoids. B
of monoid generators
is any set of generators for
S
as a
B cC . Statement
2.2 and 2.3.
(1)
To prove (2)
of elements of is generated by
S that
are
follows at once from Theorems , we
note thatthe set
less
than
Kd
is finite, and
u (Kd,(K + l)d,...,(2K - 1)d } .
The fact that each submonoid of
Zq
is finitely generated
S
14
implies,
in the usual way,
satisfied in
for
S .
B = {b^} S
(3) , let
S .
Clearly
If .
a.c.c.
on submonoids is
.
To prove ment of
that
{b^}
b^ b^
be the smallest positive ele belongs to each generating set
generates
Otherwise,
not in the monoid
let
b^
^
as a monoid,
n
since
Zq
take
B = {b1 ,b2 ,...,bn > .
b^
.
Again it
for any generating set
Continuing this process, we obtain some
then we take
be the smallest element of
generated by
{b^,b2) c c
is clear that
S
satisfies
C
C
of
^ = S
a.c.c.
on
submonoids,
S .
for and we
We have already observed that each nonzero numerical monoid is isomorphic to a primitive one.
The next result
shows that distinct primitive numerical monoids are not isomorphic.
If
and
a homomorphism of S2
S2
are semigroups, then as usual,
into
S2
is a mapping of
into
that preserves the semigroup operation. THEOREM 2.5.
Assume that
numerical monoids.
If
S
and T
h : S -- ► T
are nonzero
is a
homomorphism, then
there exists a nonnegative rational number h(s) = qs
for each
s e S .
or it maps each element of primitive and
h
S
Thus
h
to
0 .
q
so that
is either injective
is an isomorphism of
If S
S
and
onto
T
T
are
, then
S = T . Proof.
Let
s
t = h Cs) , and let it is clear that h ( s s ’) = sh(s')
be a nonzero element of
S , let
q = t/s . Take
s' e S
.
If
s'
* 0 , then
h ( s ’) =
qs '. Thus
h(s') = 0 = = s fh(s)
so
qs1 . that
If
s’ = 0 ,
15 h
is injective or the zero mapping, according as
q = 0 .
and
T
are primitive and
his surjective,
then primitivity of
S
and the inclusion
qS
that
If
q
S
q * 0
is an integer, and primitivity of
equality
T = qS
show that
S .and
T
implies equality,
If
S
and
B
any
B
°of with
Zq
has
S .
rank
self, need not be finitely generated. (l,k) t °
k
Zq
If
S
For example, for any
k . to
is a submonoid of
Z
then
S
containing is a subgroup
Z . Proof.
and
Zq
group of integers.
both positive and negative integers, of
for
with it
It is an easy step to pass from submonoids of
THEOREM 2.6.
The
Z q , we note thatsub
ZQ© ZQ , the external direct sum of
submonoids of the additive
or
d2 and
Let
= S n Zq
and
are the greatest common S2
S 2 = S n (-Zq)
•
If
d^
divisors of the elements of
, respectively, then since
ZQ
and
-Zq
are
isomorphic, Theorem 2.4 implies that there exist positive integers and
and e ^2
We show that There exist and since integer
K2
such that
^0r eac^
kd^ e
m > K2 .
S = dZ , the inclusion integers
x
and
Let
for each
k ^
d = g .c .d .{d^,d 2 } .
S g dZ
y such that
being clear. d = xd^ + y d 2
xd^ + y d 2 = (x + rd2)d^ + (y - rd^)d2
for any
r , we can assume without loss of generality that
,
16 x > Kx
and
-y >
shows that
.
-d e S
Thus
d e S
as well.
and a similar argument
Therefore
dZ c
S
Theorem 2.6 can be extended to submonoids group
Q .
Before stating this result
and
(Theorem 2.9), however,
material will be referred to again in Chapter 3. x -- ► tx
zero element
is an automorphism of
t e Q .
Q
Recall that a group
H
H
lowest terms.
If
Q by
generated by 1/m .
group of
for each
G
Q
inclusion
H
E
has p ro
is cyclic. be a finite subset of i
and each
i-s i-n
{1} u {a^/b^}^ Q
is cyclic and is generated
is locally cyclic.
In this proof, we use
(Y) to
generated by the subset
Y
denote the sub
of
(1,a^/b^,...,an/bn ) c (1/m) is
Q .
n
write
y
1 = a^x + b^y
for integers
l/b1 = x(a1/b 1) + y e (l,a1/b1) . and write
x
and
The
clear. To prove
the reverse inclusion, we use induction on
gcd{b1 ,b2)
contains
m = lcm{b^,...,bn } , then the subgroup of
In particular,
Proof.
H
Let {a./b.}? , i' i i = l
b^ > 0
Q
is locally cyclic if each
finitely generated subgroup of
Q\ {0 } , where
of
is said to have property
In particular,
THEOREM 2.7.
G
Z ,and in consider
locally if each finitely generated subgroup of E .
First, the
for each non
ing such subgroups, we frequently assume that
perty
Q ; this
Thus, each nonzero subgroup
is isomorphic to a subgroup containing
Z .
.
of theadditive
we review some of the basic theory of subgroups of
mapping
S = dZ
.
If
n = 1 , .
Then
Ifn = 2 , let d =
d = b^x + b 2y .Then
1/m = d/b^b2 =
x(l/b2) + y (1/b^) e (1 ,a^/b^,a 2/b 2) , so the equality (1/m) = (ljaj/b^,...,an /b )
holds
forn = 1
or
2 ..
17
Assume the result
for
k + 1 , let
lcm{b^, . . . ,b^}
The cases
m’ = n = k
n = k , and in the case where
and
n = 2
so that
n =
m = lcm{m',t^+^}
.
yield the desired conclusion:
1/m e (l/m’,ak+1/bk+1) £ (l.aj/bj,...,ak+1/bk+1) . An arbitrary finitely generated subgroup contained in
({1}
u Y) .
Since
and since a subgroup of a cyclic that
(Y)
is cyclic and
Q
(Y)
of
the latter group is
cyclic
is locally cyclic.
Assume that
(1)
union of an ascending sequence of
is the
is
group is cyclic, it follows
COROLLARY 2.8. H
Q
H
is a subgroup of
Q . cyclic
groups. (2) set
If H contains
{1/pk | p is prime, Proof.
ments of
(1):
H .
Let
If
Statement
k >
H
(2)
the
1/p^ e H} .
an enumeration of the
c..., and
for each H =
ele
i , then each .
follows immediately from Theorem 2.7. If
both positive and
is generated by
0 , and
{ h ^ b e
c
THEOREM 2.9.
S
is a submonoid of
negative rationals, then
Q containing S is a subgroup
Q . Proof.
zero
then
= (h^,...,h^)
is cyclic,
of
Z ,
It suffices to show that
s e S . Choose
subgroup
G
of
that
g
then
mn < 0 .
Q
t e S
such that
generated by
is a generator of
-s e S
G .
{s,t} If
group (we're using the fact that
G
st < 0 .
no n
The
is cyclic; assume
s = mg
Thus, the subsemigroup
for each
<s,t>
and of
t = ng , G
is a
admits only two total
18 orders —
one in which
— g > 0 .)
g > 0
and a second in which
-s e <s,t> c S .
It follows that
Because of its connection with numerical semigroups, we conclude this section with a theorem concerning the multipli cative semigroup of a commutative ring. semigroups, noted
If
A
and
B
then the external direct sum of
A
and
B , de
A © B , is the set
A x B , with operation defined by
the coordinatewise operations on THEOREM 2.10. (Z + , 0
and
Let
S
(Z+ ,+ ) .
A
and
B .
be the direct sum ofthe semigroups
If
his a homomorphism
a finitely generated commutative semigroup exist
m,n,a e Z+
Proof.
are
with
m * n
Assume that
into
T , then there
such that
T =
of S
h(m,a) = h(n,a) .
.
For each of the lr
first For
k + 2
primes
p^ ,
1 < j < k + 2 , let
we write
v.
be the
h(pj,l) = E ^ a ^ t ^
•
(k + 1)— tuple
k+ 2 a . Thus, the set {v.}._, is linearly deJ1 JK J J pendent over the rational field Q . An equation of linear (1, a
depencence of EbjVj and
{v^. }
= ^cjvj bj
* c^
=
Eb.v. 3
3
> where b^ for some
implies that n =
over
Qcan be written in and
j .
Z b . = Ec. = a . •Then
c^.
for some
3
integers
s^jvj
k+2 k-; m = II. np.
hfm.a) = I ^ J h f p ^ ' . b ^
= Ec.v. = h(n,a) ; moreover, 3
are positive
The equality Let
the form
= ^cjvj and let
= E*+2b jh(pj ,1) =
m * n
since
b. * c. 33
’’’
j .
THEOREM 2.11. commutative ring
If the multiplicative semigroup of the R
is finitely generated,
then
R
is
19
finite. Proof. acteristic
Consider first the case where p .
In this case we show
finite for each
x
The ring
vector space over
, the field with
assumption that
R
prime char
that
is (R,#)
is
can be considered as a
linearly
leads to a contradiction, R
has
e R ;this is sufficient since
finitely generated.
of
R
for in that
p
elements.
independent
The over
case the subspace
V
spanned by
{x1 }” is isomorphic as a ring to the ring
XZp[X]
, where
is an indeterminate over Z^ .
{P^}?= ^
is a complete set of nonassociate prime elements of ei e-\r distinct from X , then the mapping (Pj ••-Pv ,a) -- ►
Zp [X]
x [P-^(x)]
ei
X
e ...[P Cx)] v
Theorem 2.10) in {x1 }”
(R,*)
is an imbedding of
x,x^,...,xU ^ .
spanned by {x^}^ * , so finite.
(as in
, contrary to that result.
is linearly dependent, so some
bination of
S
But if
V
xu
Thus
is a linear com
It follows easily that
is finite and
{x1 }*
is
is also
This completes the proof in the case where
prime characteristic.
V
R
has
The rest of the proof amounts to a
reduction to that case. In the general case, let generators for
(R,*)
x € R , the mapping into
and let
be as above.
For any
is a homomorphism of
S
(R,#) , and hence Theorem 2.10 shows that for some and
has finite additive order. and
S
(m,a) -- ► mxa
m,n,a e Z + , mxa = nxa
K
a finite set of
b
so that
the ideal of
R
Kx^ = 0
m * n .
Thus, a power of
x
We choose the positive integers for
1 < i < k .
consisting of all elements
Let
r e R
I
be
such that
20 Kr = 0 .
The residue class ring hi
finitely many monomials that
h^ < b
for each
i.
R/I hk
is finite since only
in
x^,...,x^
Therefore
R/I
characteristic, and the definition of
I
are such
has nonzero
implies that
R
el ev also has nonzero characteristic c . Let c = p^ ...pv be ®i the prime factorization of c . If R^ = {r e R| p .Jr = 0} for
1 < j < v , then
R,,R~,...,R , and 1 2 v As
Rj
R
R.
j
is the direct sum of the ideals e. has characteristic p .-1 for each vj
is a homomorphic image of
semigroup is finitely generated. finite,
j . J
R , its multiplicative Thus,
it suffices to prove that each
to prove that R^
R
is finite.
is We
therefore assume without loss of generality that the charac teristic of
R
is a prime power
pe .
Then
R > pR >...> p e *R > p eR = (0) , and the additive group of p*R/p*+ ^R
is a homomorphic image of that of
i .
if
Thus,
R/pR
R/pR
is finite, then
is a ring of characteristic
image of R/pR
R .
R
p
R/pR
is finite.
for each Now
which is a homomorphic
By the first case of the theorem considered,
is finite and this completes the proof of the theorem. Section 2 Remarks If
B
is a subset of a monoid
of the submonoid of
S
S , then the definition
generated by
in the proof of Theorem 2.4.
B
appears implicitly
The term is defined, of coiirse,
as the intersection of the family of submonoids of taining
B u {0} .
It is denoted by
to adjoining the zero element of S
generated by
S
^
S
con
, for it amounts
to the subsemigroup of
B .
Additional material on numerical semigroups can be found
21 in [112, Sect.
38],
[75], and [71].
In particular,
there has
been some work on determining the smallest positive integer K
such that
K + Zq
merical monoid b , the value
S .
is contained in a given primitive n u For
S
of rank 2 generated by
K = (a - 1 ) (b - 1)
not, in general, be improved.
a
and
given in Theorem 2.2 can
Much of the interest in n u
merical monoids in commutative algebra stems from the study of subrings of the polynomial or power series ring in an in determinate {t
ni k
t
over
R
that are generated over
°f pure monomials.
exponents come from the monoid
of
the ring
^
over
R[t
nl
R .
by a set
The ring in question is just
the set of all polynomials or power series over
polynomials,
R
^
,...,t
nk
]
R
whose
; in the case of
is the semigroup ring
Two recent papers dealing with
rings of this kind are [22] and [49]; polynomial rings are treated in [22], while [49] considers power series. Theorem 2.8 was proved by Isbell, and our proof is merely an elaboration of the proof given in [79].
The question of
whether Theorem 2.8 extends to noncommutative rings appar ently remains open.
22
§3.
Ordered Semigroups
An examination of the proofs in Section 2 shows that the order relation on the results.
Z
plays a significant role in several of
In this section we determine the class of
abelian semigroups that admit a relation of total order com patible with the semigroup operation.
First we need some
definitions. A relation
~
on a semigroup
(S,+)
is said to be
compatible with the
semigroup operation if
a + x ~ b
+x
a,b,x e S .
trary set
X
transitive,
for
A relation
and asymmetric
(~
is asymmetric if
a = b ) , and a partial order
total
order on X if for distinct elements
either
a ~ b
mean
~ on
implies an arbi
is called a partial order if it is reflexive,
b ~ a imply that
noted by
a ~b
or b ~ a .
b
and
~
is
a
e
X ,
A partial order is frequently de
< and the notation
b < a and
a,b
a ~ b
* a .
a > b
or
b
< a is
used to
To say that a semigroup
S
is
partially ordered (respectively, totally ordered) under a relation
0 a 1 a
for each a)
is a
=
26 positive subset of {a } ^ (b } a a
if
a
dinal order on G
G . a
G .
> b
The induced partial order, defined by for each
a
One extension of the cardinal order on
to a total order is the lexicographic order on
fined by well-ordering the set if a
a , is called the car* ---
a
a > b a0 a0 * b
.
a
A , then setting
for the first element
an 0
The external weak direct sum
family
is a subgroup of
G , de
of
A
for which
G„ = Ew .Z 0 aeA a
G , and the orders on
of the Gq
in
duced by the cardinal and lexicographic orders,respectively, are called the cardinal and lexicographic orders on Another total order on
Gq
in frequent use is the reverse
lexicographic order, whose positive cone consists of the nonzero elements of positive.
Z
0
and
whose last nonzero coordinate is
We note that the assumption that each
morphic to above;
Gq
Gq .
plays only a minor
Z^
is iso
role in the definitions
the definitions are meaningful for any family
of totally ordered semigroups. is isomorphic to
It is the case where each
Z^
Z , however, that is used in the next
corollary.
COROLLARY 3.3.
If
a ^ree subset of a
torsion-free abelian group
G , then there exists a total
order
0 < ga
^
on
Proof. (g 6a }
G
such that
for each
The subgroup
is isomorphic to ^
G
Ew aeAAZ a 9, where
Extending the cardinal order on &
E
a
gene**ated by
Za a Z
.(g ) aeA &a
.
for each
a .
to a total order
yields the desired conclusion. For the sake of future reference, we state formally the
27 result proved concerning orderability of a commutative semi group . COROLLARY 3.4.
The abelian semigroup
S
admits a total
order compatible with its semigroup operation if and only if S
is torsion-free and cancellative. Also for the sake of reference, we record one consequence
of the proof of Theorem 3.2. COROLLARY 3.5.
Assume that
the torsion-free group nx k Pq
for each
positive subset and
P
G
and that
n e Z+ . of
G
PQ
Then
is a positive subset of x e G
Pq
such that
is such that
can be extended to a P
satisfies
(iii)
—x e P . Section 3 Remarks The result
totally ordered
that atorsion— free abelian group can be is due toLevi
[92], but the method used in
the proof of Theorem 3.2 is closer to that of [80]; the class of lattice— ordered abelian groups is also discussed in [80]. For
additional information concerning the question of what
nonabelian groups can be totally ordered, see Chapter 3 of [46].
In particular, a (multiplicative) group with the p r o
perty that
x * y
implies
x11 * y11
for all
n e Z+
need
not admit a total ordering compatible with the group operation [93].
28
§4. If
S
Congruences
is a semigroup,
the homomorphisms defined on
and the homomorphic images of determining the structure of
S
play an important role in
S .
An equivalent and con
venient way of considering homomorphisms on the notion of a congruence on congruence on
S
S
S
is through
S , defined as follows.
is an equivalence relation on
compatible with the semigroup operation.
S
A
that is
Theorem 4.1 states
the basic relationships between homomorphisms and congruences. The proof of Theorem 4.1 is routine and will be omitted. THEOREM (1) note by s
4.1.
If
~
Let
S
is a congruence on
eS , and let
(2)
S/p
under
de
~ foreach
S/~
is an
[a] + [b]= [a + b]
defined
by f(s) =
S/~ ; moreover,
[s]
, is
f(s^) = f(s2)
si ~ s 2 * h : S -- ► T
T , define the relation
[s] -- ► h(s) s e S
onto
Conversely, if
h(a) = h(b) groups
S
s e S,
under
Then
the operation
f : S -► S/~
a homomorphism of and only if
of s
S/~ = {[s ] | s e S} .
and the mapping
onto
S , then for
[s]the equivalence class
abelian semigroup under
S
be an additive semigroup.
.
Then
and
T
, where
p
is a homomorphism of on
is a congruence
p
S
by
on S
a p b and
if
thesemi
are isomorphic under the mapping [s]
denotes the equivalence class of
p .
The semigroup
S/~
defined in
called the factor semigroup of
S
(1)
of Theorem 4.1 is
with respect to
Theorem 4.1 shows that, to within isomorphism, semigroups represent all homomorphic images of
~ .
these factor S .
if
29
If
and
are congruences, then we write
~2
if
a
b
implies a ~ 2 b
and
~2
as
subsets of
and
only of
f°r
a >b €
S , the relation
is contained in
form of the relation
< ~2
Thus,
S .
is the assertion that the onto
S
is
the smallest congruence on
gruences on family
S
S
S
is well
are related, and
is the identity congruence
The intersection of any family of con
is again a congruence on
^~a ^aeA
S/~ 2
S * S , the universal con
gruence under which any two elements of
.
is a partial
is well defined, it is a homomorphism).
The largest congruence on
i = {(s,s) | s e S}
or lub{~a)) and a greatest lower bound
(denoted
g.l.b.{~a > or glb{~a >) in the set of congruences on
S .
The relation
means that
a ~
a
~ = glb{~a > b
note that any subset
for p
each of
iseasy to describe: a.To describe
S x S
lub{~
a
} , we
generates a congruence on
S , namely, the intersection of all congruences on contain
p .
lub{~a )
is the congruence generated by U
of lub{~a >
a ~ b
S
that
We proceed to describe this congruence; since
follows.
The congruence
~
, a description generated by
p
is
constructed in three steps. 1. and
i
tains and
Set
Pq = p u p_1 u i , where
is the identity congruences on p
S .
Then
and is both reflexive and symmetric.
pQgenerate the same congruece on 2.
p 1 = { (b ,a)| (a,b) e p}
Set
Pq
con
Clearly
p
S .
p1 = p0 u {(a + c,b+ c) | (a,b)
e pQ
and c e
S} .
30 Then
is reflexive, symmetric,
semigroup operation on { (a + c,b + c) | (a,b)
S
.
IfS
and is compatible with the is a
monoid, then
p-^ =
e p and c e S} since pn is 0 u Again it is clear that Pq and
tained in this set.
conp^
generate the same congruence. 3.
Let
such that
~ =
((a,b) |
a = an , b 0
= a^, and t
i = 0,1,...,t - 1} . rated by
there exist
aQ,a^,...,a^
fa. ,a. .. ) e p i i+I 1
Then ~
is the
for
congruence on
S
gene
p . P =U ~ is the union of a family of cona a 7
In case gruences on
S , thenp = pQ =
need be applied to A congruence
p ~
p^ , and only Step3 above
in order to obtain on
S
there exists a finite subset
that
is the
~
Noetherian
S
P
of
congruence generated by
if each congruence on
As usual, each congruence on only if
lub{~a )
satisfies
a.c.c.
S
S
p
S
S
is finite.
such
, andS
is
is finitely generated if and on congruences, which is true < ~2 < * * *
In Section 7 we prove that a
is Noetherian if and only if
rated as a semigroup.
S x S
is finitely generated.
if an only if each strictly increasing set congruences on
.
is said to be finitely gene
rated if
monoid
€ S
S
is finitely gene
We return briefly to a consideration
of least upper bounds and greatest lower bounds of congruences in Section 5.
The rest of this section is devoted to an
examination of some examples of congruences that naturally arise in the remainder of the text. THEOREM 4.2.
Assume that
T
is a subsemigroup of the on
semigroup
S .
Define a relation
~
a + t = b
+ t
for some t e T .
Then
S
by
a ~ b
~is a congruence
if
31 on
S
and
If
p
is any congruence on S
S/p
[t]
is cancellative in such
S/~
for each
that the image of
consists of cancellative elements of Proof.
a ~ b
and
for some
Clearly
~
t1 »t2
6 T
>
so
c.
b +
s + t^
If
[a] + [t] = [b] + [t]
then a ~ b
a + ft2 +
a + t^ = b + t^ s e S
so that
for each
[a] = [b] . t e T .
p > ~. If
b + t2 = c + t2
= c + Ct2 + ti^
and
a + s + t^ =
~
congruence on
is a
a,b e s
for some
Thus,
and
in
implies
for some
a + t + t ^ = b + t + t ^ and
S/p , then
a + t^ = b + t^
a ~
for each
T
is reflexive and symmetric.
b ~ c , then
Moreover,
t e T .
[t]
and
t^ e t
t e t
~ < p
,
, and hence
is cancellative in
The assertion that
S .
for
p
S/~
as de
scribed is clear. In the case where S
~
defined on
as in Theorem 4.2 is called the cancellative congruence on
S . T
T = S , the congruence
For a subsemigroup is cancellative in
respect to semigroup
T T
T
of
S
such that each element of
S , the quotient monoid of
was defined in Section 1. , the quotient monoid of
{[t ] | t e T}
S
defined in Theorem 4.2.
, where
with
For a general sub with respect to
is defined to be the quotient monoid of the subsemigroup
S
S/~ ~
T
with respect to
is the congruence
The definition is reminiscent of the
definition of the quotient ring
of a commutative ring
with respect to a multiplicative system
R
N , and indeed there
is a connection between the two, as the next result shows. THEOREM 4.3.
Assume that
in the commutative ring under multiplication,
R .
N
is a multiplicative system
Regarding
R
the quotient monoid of
as a semigroup R
with respect
32 to its subsemigroup the ring
N
is the multiplicative semigroup of
.
Proof. We recall the definition of R . Let I be ----N the ideal of R consisting of all r 6 R such that rn = 0 for some from
R
n e N
and let
onto R/I .
system in (R)
<j> be the natural homomorphism
Then
and
(N)
is a regular multiplicative
is defined to be N = (<J>0)/<J> (n) | r e R,n e N} . The elements of the
<J>[NJ quotient monoid of in the form
R
R
[r]/[n]
, where
x e R
with repect to
a ~ b
if
an = bn
we show that
for some
yn = xn iff
The next result
N
can be represented
denotes the class of
~
is defined on
n e N .
for each for some
y - x e I
[x] = x + I = <J>(x) ,
[x]
~ , and
[x] = <J>(x)
if and only if (y - x)n = 0
with respect to
by
To complete the proof,
x e R .
Thus,
n e N .
iff
R
But
y e x + I .
y e [x]
yn = xn
iff
Therefore
and this completes the proof. is essentially contained inSection
We restate it here in the language of congruences.
1.
The
proof should be familiar from group theory, and hence it is omitted. THEOREM 4.4. is a subgroup of a ~ b
Assume that S
containing
to mean that
is a congruence on
S
a = b + h
is a monoid and that 0 .
For
for some
a,b e S, define h e H
S , [a] = a + H , and if
of all invertible elements of vertible element of We remark that if
S , then
H
[0]
H
. Then
~
is the group
is the only in
S/~ . S is a group, then the only congruences
33 on
S
arise as in Theorem 4.4 —
that is, as congruences with
respect
to a subgroup of S .
gruence
on the group
S and
let H
= {s e S | s
H
subgroup of
S ,for
a ~0
,b ~ 0imply that
b so
~ 0 ; also,
is a
a + b ~
a + b
To see this,
Thus
H is asubgroup of
S and
a - b
£H
.
iff a 6 b + H
There is an analogue,
a
let
-a ~ 0 a ~ b
~
be a con
~ 0} .Then
- a
iff
so
0~
a - b~
- a.
0
iff
for torsion-freeness, of the can
cellative property in Theorem 4.2.
We can kill more than the
torsion-free bird, however, with a single stone. THEOREM 4.5. and that tegers.
S
is an additive semigroup
is a multiplicative semigroup of positive in
For
for some m[a]
M
Assume that
a,b e S , define
m e M .
= m[b]
Then
for some
~
a ~ b
to mean that
is a congruence on
[a],[b]
e
S/~
ma = mb
S , and if
and some
m e M , then
[a] = [b] . Proof.
The reflexive and symmetric properties of
are immediate.
If
m^a - m^b
andm^b = m^c
mm a = in m e and m in e M ,so 12 12 12 Moreover, m^a = m^b implies that for
x e S ; hence ~
m[a]
= m[b]
means
nm e M , it follows that In case
M = Z+
torsion-free and such that
S/p
~
m^(a + x) = m^(b + x)
nma = nmb a ~ b
, then
~is alsotransitive.
is a congruence on that
S .
for some
and
The equality ne M .
Since
[a] = [b] .
in Theorem 4.5, the semigroup is the smallest congruence
is torsion-free.
~
p
S/~ on
is S
In considering nilpotent
elements of a semigroup ring over a ring of prime character istic
p
in Section 9, we shall encounter the case of
34
Theorem 4.5 where
M = {P 1}i=0
iS th° S6t °f powers of p ;
the congruence is denoted by
~ in
ferred to as
If ~
p— equivalence.
congruence on
S , we say that
S
this case, and is
re
is the identity is
p— tors ion-free.
Another congruence encountered in Section 9 is that of asymptotic equivalence, defined by setting a ~ b if there + exists K e Z such that ka = kb for each k ^ K . Theorem 4.6
provides an alternate way of viewing this re
lation, and Theorem 4.7 is the analogue, equivalence, of Theorem 4.5.
for asymptotic
We say that
S
asymptotic torsion if distinct elements of
is free of S
are not
asymptotically equivalent. THEOREM 4.6. na = nb
Assume that
for some
a,b € S
n e Z + . Let
positive integers
n .
additive semigroup
Then
M
M is
are such that
be the set ofall such a subsemigroup of
the
Z+ , and the following conditions are
equivalent. Cl)
a
and
b
are asymptotically equivalent.
(2)
M
is primitive.
(3)
T h e r e exist
n ^ , n 2 , .. . ,n^ e
M
such that
g.c.d. {nj_,. . .n^} = 1 .
(4)
There exist
m,n e M such that
m
and
n
are
relatively prime. Proof. Conditions
That (1)
M and
is closed under addition is immediate. (2)
are equivalent by the definition
of asymptotic equivalence, and the equivalence of and
(4)
(2) , (3) ,
follows from Theorems 2.2 and 2.3 and from the de
finition of primitivity.
35 THEOREM 4.7.
Assume that
~
is
the relation of
~
is
a congruence on
asymptotic torsion. that
S/p
S
asymptotic equivalence on S , and the semigroup If
p
is any congruence on
We prove only that
torsion.
Thus, assume that
[a]
[b]
and
there
exists
such that
k 2 €Z+
[b]
in
S/~
m,n
k 2na
= k 2nb .
it follows
[a] = [b]
and
S/A
Since
such
Then there ma ~
mb
and
k^ e Z+
such
By the
same token,
relatively prime to k^m
form (4.6) that S/~
Does there
k2
S
are such that
so that
g.c.d.{k^,n} = 1 . , with
free of
p > ~ .
Theorem 4.6 shows that there exists
k^ma = k^mb and
Then
is free of asymptotic
are asymptotically equivalent.
that
that
S/~
[a],
exist relatively prime integers na ~ nb .
S. S/~ is
is free of asymptotic torsion, then
Proof.
prime,
is a semigroup and that
andk2n
k^m
,
are relatively
a ~ b .
Therefore
is free of asymptotic
torsion.
exist a minimal congruence A
is torsion-free and cancellative?
on
S such
The answer is
affirmative, and the congruence in question is, in this case, the least upper bound of the cancellative congruence on and the minimal torsion-free.
congruence
A
on
Rather than regard
bound, however, gruence
p
S A
it's easier to define
such that as this
S
S/p is least
it directly.
upper
The con
will reappear in Section 8 , where zero divisors
of semigroup rings are considered. THEOREM 4.8. S
as follows:
such that
na +
Define the relation a A b if there exists x = nb + x .
Then
and is the smallest congruence on
A n
A is S
on the semigroup e Z+ and
x e S
a congruence on
such that
S/A
is
S
36 torsion-free and cancellative. Proof. a A a
since
nb + x and
We verify all the details for a change. a
+ x=a + x
implies
for each
nb +x = na + x .
If
n^a
na
+ x =
+ x^ = n^b + x^
n 2b + x 2 = n 2c + x 2 , then it is easy to check that
n ln 2a + n 2x l + n lx 2
=n ln 2c + n 2x l + n lx 2 *
na + x = nb + ximplies that for each
y e S .
in S/A , then
for some
and
m
Similarly,
and
x . S/p
m
[a] = [b] .
na A nb
and
if
= n(b + y) + x
a Ab
and
means that
[a]
= [b] .
m(a + c) + x =
S/A
is torsion-free and
is torsion-free and can x = nb + x
for
Hence n[a]p + [x]p = n t^]p + tx ]p
is also torsion-free,
If
so mna + x = mnb + x
na +
is cancellative,
S .
x ; again this implies that
S/p
a A b , then
Moreover,
is a congruence on
Therefore
Finally,
cellative and if
Since
A
x , implying that
for some
cancellative.
and
+ y) + x
[a] + [c] = [b] + [c]
m(b + c) + x a A b
n(a
Therefore
n[a] = n[b]
n
x e S , and
Thus
some
in
s /p
n[a]p = n[b]p , and since
[a ]p = [b]p .
Therefore
• it
a p b
and
this completes the proof. We consider one other class of congruences on a semi group S .
S
before ending this section.
The relation
either
a = b
or
on
S
where
can be thought of as x+°° = «>+x
= «>
x,y e S\I , their sum in
be an ideal of
is a congruence on
called the Rees congruence modulo
S/I
I
defined by setting
a,b e l
is sometimes denoted by
Let
S/I S\I for S/I
I .
a ~ b
if
S ; it is
The factor semigroup
instead of
S/~ . Setwise,
, together with a symbol each
x e S/I
is either
and where,
x + y
or
00 ,
00 for
,
37 depending upon whether
x + y e S\I
factor semigroups
have some of the properties that one
S/I
or
x + y e I .
The
would anticipate of them from ring-theoretic notation. example,
if
For
{Ja }
is
the set of ideals of
then
Ja -- ► Ja/I
is
an inclusion-preserving bisection from
{Ja )
onto the set of ideals of
S/Ja ~ (S/I)/(Ja/I)
for each
S containing
I ,
S/I , and a .
This result can be used
to show that a Noetherian semigroup satisfies the ascending chain condition for ideals, but we give a proof based on first principles in the next section. A straightforward description of where
~
is a Rees
congruence,
lub{p,~}
, in the case
is contained in
the next re
sult. THEOREM 4.9.Assume that the
semigroup
the ideal
I
there exist Then
S , where of
and
~
are congruences
y
be the relation a p c , c ~ d,
{(a,b) | and
d p b}
and
It is clear that
b y e ,
y
is reflexive, symmetric, and
then there exist
w,x,y,z e S
a
p w, w ~ x., x p b , b p y , y ~ z ,
y
p z , then the transitive property of
nition of and
y
y * z
therefore
imply that so
w ~ x
w ~ z
gruence on y < lub{p,~)
S .
.
.
that it is compatible with the semigroup operation. a y b
on
is the Rees congruence modulo
e S such that
y = lub{p,~} Proof.
~
S . Let
c,d
p
a y e .
and
and again
and
z p c . p
y ~ z imply that
Since the relations
Thus
so that
If w p x
case,
w* x
w,x,y,z £ y
I ;
is a con
p < y , ~ < y ,
are clear, it follows that
or
and the defi
In the other
a y e .
If
y = lub{p,~}
and .
38 Section 4 Remarks It is traditional to blur the distinction between con sideration of a relation A x A
p
on a set
A
as a subset of
or as a rule by which certain pairs of elements are
related.
Considered as a subset of
A x A , we write
p c A x A ; considered as a rule, we write
a p b .
We have
maintained this dual consideration of congruences in Section n 4 and will continue to do so. If is a finite set of relations on a *?=^ p ^
A , then
glb{p^}^
is frequently denoted by
; considered as a subset of
A x A , we have
a1? p. = n? p . The corresponding lattice-theoretic noi=l l i=l i „ n n tation for lub{p^>1 is vi=iPi • For relations, n n v, p. is u,p. as a subset of A xA ,but whereas the 1 i 1 l ’ finite intersection of congruences on
a semigroup is a con
gruence, the set-theoretic union of finitely many congruences need not be transitive.
39 §5.
Noetherian Semigroups
Recall that a semigroup
S
is Noetherian if the ascend
ing chain condition for congruences holds in main purpose
S .
While the
of this section is to prove Theorem 5.10, which
shows that a Noetherian monoid is finitely generated, the material divides rather naturally into three related parts. Some basic consequences of the developed in the first part.
a.c.c.
on congruences are
The second part includes
several new concepts and is devoted to proving Theorem 5.8, a form of Primary Decomposition Theorem for congruences on a Noetherian semigroup.
Theorem 5.10 and its proof constitute
the third part of the section. THEOREM 5.1. (1)
The
each ideal (2)
be a Noetherian semigroup.
for ideals of
S
is satisfied.
Hence
is finitely generated. S
is a monoid and if
vertible elements of G
S
a.c.c.
ofS If
Let
S , then the
G
is the group of in
a.c.c.
for subgroups of
is satisfied. (3)
PC = S\P (4) then
A
If
P
is a proper prime ideal of
S , then
is Noetherian. If
A
is an ideal of
S
such that
A
is a monoid,
is Noetherian.
Proof. Let I cl - R 0 [X;S0] ; i C E r ^ i )
t
,
,
= Ey (ri)X('b ') = 0 follows that
I E ker 4>* .
take a nonzero element to see that and
A[X;S] + I ; t
are surjective.
t
jectivity of these is clear.
m > 1
<J>*
is surjective.
homomorphisms, as well as the statements concerning
y*
is
is surjective.
is surjective if Proof.
; y*
is a homomorphism and its kernel is the ideal
is surjective if (3)
A[X;S]
.
fj = -f2 , so f
if
is the
sur-
kernel
I = ker <J>*.
Since
* .
m = 2 , then clearly
= fjX*51 - fxX S2 € I .
there exist distinct elements
si»sj
°f
I t ’s easy (s^) =
Cs 2 D
m > 2
,
For
Supp(f)
such that
70
* By induction, 1= ker * . and
Thus,
(r^XS i — r^XSj)
with fewer than g
is in
m
is an ele
elements in its support.
I , and hence so is
The assertion concerning
f .
ker t
Therefore
follows from
(1)
(2) . COROLLARY 7.3.
R
g = f —
and that
I = ({rXa —
Assume that
~ is
A
is an ideal of the ring
a congruence on thesemigroup | re
rX^
R
anda ~ b})
,
, and
R [ X ;S]/ (A[X;S]+I) ~ (R/A)[X;S/~] The ideal
I
Theorem 7.2 shows that
If the ring
.
of Corollary 7.3 is called the kernel
ideal of the congruence
elements of
. Let
.Then
R[X;S]/A[X;S] ~ (R/A)[X;S] R [ X ;S ]/1 ~ R[X;S/~]
S
~ . I
a form
the
The proof of part
(2)
of
consists of all finite sums of b
rX
R isunitary,
— rX
then
course, a set of generators for
, where
r e
Randa
(xa — X^ |a ~ b> is,
I .
~b.
of
Ideals of the semi
group ring that are kernel ideals have some interesting properties, as we proceed to show. THEOREM 7.4. the ring
R
and
on the semigroup (i) Moreover, fied.
Assume that I
A,
is the kernel ideal of the congruence
S .
nAeA{A A [X;S]+I} = ( n ^ ) if
R
are ideals of
is unitary,
then
[X;S] + I . (2) and
(3)
are satis
~
71 A [X;S] n (B[X;S]+I) = (A n
(3)
Proof.
(1):
homomorphism
of
to theassertion thisequality (2):
Since
I
R[X;S] that
B)[X;S]
+ AI.
is the kernel of
onto
the
[R X;S/~] , (1)
canonical
is equivalent
n(A^[X;S/~]) = (nA^)[X;S/~]
,and
is clearly satisfied.
Since
R
is unitary,
A[X;S] = A*R[X;S]
Thus
(A[X;S])•I = AI .
The inclusion
patent.
of the reverse inclusion is similar to the
The proof
proof of
(2)
in Theorem 7.2.
f = ^i=i^ix S ^ 6 1 n A[X;S] and
m = 2
implies
.
AI
.
Thus, Then
distinct
s ± > sj
moreover,
e A [ X ;S]
implies that
(3):
e AI
In view
f^ e
f e AI .
A
For
€ Supp(f) such that
, we have of
m > 1 , and m > 2 , si ~ sj >
f^ e A .The induction
s* s• f — f^(X 1 — X J) e AI , and since
hypothesis implies that X 5^
implies
f = f^CX5! — Xs2) > where
there exists
f^(XS;*-
take a nonzero element
f eI
s^ ~ s^ , from which it follows that
f
£ (A[X;S]) nI is
f e AI
(1) and
.
(2) , assertion
(3)amounts
to saying that intersection distributes over sum in This is known to hold if
B c A .
(3) .
But we can reduce to that
case since A [ X ;S] n (B[X;S]+I)
=
A [X;S] n (A[X;S]+I)
n(B[X;S]+I)
A [ X ;S] n { (A n B)[X;S]+I>
=
.
This completes the proof of Theorem 7.4. Theorem 7.5 establishes a close relationship between the lattice of congruences on
S
and the set of kernel ideals
72 of
R[X;S]
and
.
In the statement of Theorem 7.5, we use
, respectively, to denote the least upper bound and
a
greatest lower bound of two congruences on THEOREM 7.5. group.
Let
With each
kernel ideal
R
congruence on
I
of
ideals of
R[X;S]
I .
S
into the lattice of
has the following properties.
is injective.
(2)
0
preserves order — that is,
p1
P2
S . (5)
The congruence
only if the ideal { (s jt.)}11 i i i=l generates I Proof. S ,
be a semi
: ~ ------ ► I(~)
The mapping 0
0
(3)
S
S ,associate the
(1)
only if
S .
be aring and let
from the lattice of congruences on
on
v
I (~)
~
is finitely generated if and
is finitely generated.
generates ~ if and only
if
If
p^ and
p2
are distinct congruences
sp^t
while
but
not in ICP2D •This proves
(s,t)
p^ < p^
{ p2 .
Then
Xs (1) .
s,t e Xt
the fact that
a {rX
I Cp -j^D •
inclusion I(p
A p 2) c
That
b - rX| ap-^b) I(p1)
S
such
is in ICP^D
is clear, and the other part
follows from The
{XSi — X**}1? i=l
•
then wemay assume that there exist
implies
In fact,
on that ,
ICP^D £ I ^P2^ of
(2)
generates
n I( p 2)
in
(3)
73 follows from
(2) .
Part
(3)
the inclusion may be proper. I (p1) + I ^p2^ - ^ pi V p2 ^
of Theorem 7.6
implies
that
The inclusion
in
^
also follows from
(2) .
For the reverse inclusion, it suffices to show that a b X - X e Ifp^) + IC P 2 ^ f°r each (a,b) e p^ v p2 . From the description of
p^ v p2
that there exist elements
given in Section 4, it follows s^,s2 ,...,sn e S
such that
a = Sj.b = sn ,s1p1s2 ,s2p2s3 ,s3p1s4 ,...,sn l p2sn .
Then
xa -
XSi+1
in
Xb = Z)"Ji(XSi -
I(p1) + I(p2) .
and the equality
XSi+1) , where each Xa -
Therefore
XSi -
Xb £ Ifp1) + K
I (p^ v p2) = I(p^) + 1 CP2)
p 2)
then follows.
It is clear from the second statement that
For the converse, note that if then the generating set
{X
a
I(~) I
-
X
i
b
s
finitely generated,
| a ~ b}
for
I(~)
proceed to prove the second statement.
generated by clear. if
~ .
Let
{XSi -
•
We define a congruence
Xs -
X^ e J
.
Clearly
each
i , ~ < y , and hence
that
I(~) = I(y)
y
y
J = Ify)
I
c
{XS i -
be the congruence on
We have just proved that
I (y)
S
is
(5) .
We
S .
R [ X ;S ]
J c I (~)
by setting Since
{XS i —
generated by is generated by
I
for
It follows
X*^}^ .
generates
is
s y t
s^ y t^
I(y) s I(~) .
X^i}^
~
be the ideal of
on
con
Thus, assume that
The inclusion
is generated by
versely, assume that let
J
I(~)
, whence
finitely generated by the second statement in
generates
~
is finitely generated.
tains a finite set of generators for
)}^
(5)
We observe that the first statement
finitely generated implies that
(Cs^t
,
established.
Assume for the moment that the second statement in has been established.
is
Con
and n {(s^,t^)}^ .
74
{XS i y = ~
X t i)1 , and hence and that
~
I fy) = I (~)
is generated by
» implying that
{(s^,t^)}^
.
Theorem 7.5 has a number of important consequences. fore looking at some of these, we restate case where on
G
S = G
is a group.
correspond to the subgroups, andG/~ , for
is unitary,
generated by 1 —
the corresponding ideal {1 -
X^ e I (~)
X*1 | h e H}
{1 -
Xa | a e A}
~
H , is I(~)
the G/H .
of
R[X;G]
ifg e H .For a subset
A generates
generates
If is
in thiscase, and
if and only
G , it followseasily that
in the
In that case, the congruences
congruence corresponding to the subgroup R
(7.5)
Be
I (~) .
H
A
of
if and only
if
These observations, to
gether with Theorem 7.5, yield the following result. THEOREM 7.6. Define a mapping
Let <J>
R
be a ring and let
G
be a group.
from the lattice of subgroups of
into the lattice of ideals of
R[X;G]
a subgroup of
is the kernel of the canonical
map from
G , then
R[X;G]
onto
(H)
R[X;G/H]
as follows.
G
. The mapping
If
H
is
has the
following properties. (1) (2) only if (3)
is injective. For subgroups
of
G , H^ c H 2
if and
(H^) c 4>(H2) . (H
n H 2) c
if and only if Moreover,
H ^ , H^
if
^(H ) n
H^ s H 2 R
or
<j>(H2) , with equality holding
H2 “ H1
is unitary, then
’
(4)
and
(5)
are satis
fied. (4)
d>(Hi
+ H 2) =
(5)
The subgroup
0^) +
<J>(H2) •
H
G
of
isfinitely generated
if
75 and only if
<J>(H) is finitely generated.
set
Ggenerates
generates
(H^ n
or
and choose
Then
g e H 2^H 1 *
= rX° -
H 2) =
requires proof. Let ^
rXh+ rXh+g -
hence in (J>(H^ n H 2) .
(H2),
hence
fact, a
.
Assume that
are not
if
Only the assertion in
n 4>(H2)
and let
H
In
n H 2 ,it follows that
Since
rXg
6 Hl is
h + g
h e H^
in
and
g
n H 2 , and
H, c H . 1 “ 2 We remark that the lattice of subgroups of the nontriv
ial abelian group
G
is linearly ordered if
and only if
G
is cyclic of prime-power order or a quasicyclic group (Theorem 19.3), and hence examples where the inclusion in (3)
is proper abound.
S x S
on
S , then
If
S/~
~
is the universal congruence
is the semigroup with only one
element and the semigroup ring R . of
R[X;S/~]
is isomorphic to
We call the corresponding homomorphism R[X;S]
kernel
onto
I
R
the augmentation map on
R[X;S]
is called the augmentation ideal of
is the ideal generated by THEOREM 7.7. monoid.
I^r^XS i -- ^ l r i
Let
R
{rXa -
rX*5 | r e R
; its
R[X;S] and
, and
a,b e S} .
be a unitary ring and let
S
be a
The following conditions are equivalent.
(1)
The monoid ring
(2)
R
Proof.
R[X;S]
is Noetherian and Assume that
is a homomorphic image of
S
R[X;S] R[X;S]
is Noetherian. is finitely generated. is Noetherian.
Since
R
, it is also Noetherian.
76 Moreover,
R[X;S]
satisfies a.c.c.
Theorem 7.5 implies that —
that is,
S
S
on congruence ideals, so
satisfies
is Noetherian.
Thus,
a.c.c. S
on congruences
is finitely gene
rated by Theorem 5.10. Conversely,
if
R
is Noetherian and
is finitely generated, then
S =
<s1 ,...,sn >)
R[X;S] = R[Xs 1 ,...,XS n ]
finitely generated ring extension of
is a
R , and hence is a
Noetherian ring. We remark that the case of Theorem 7.7 where
S
cancellative is much easier than Theorem 7.7 itself.
is This is
so because the proof that a cancellative Noetherian monoid is finitely generated (Corollary 5.4) is less profound than the proof of the general case
(Theorem 5.10).
The next result is
the converse of Theorem 5.10. THEOREM 7.8. Proof.
Let
monoid ring
A finitely generated monoid is Noetherian. S
be a finitely generated monoid.
Z[X;S]
The
is Noetherian by Theorem 7.7, and this
implies, by Theorem 7.5, that
S
satisfies
a.c.c.
on
congruences. A monoid
S
is said to be finitely presented
finitely definable) if rated free monoid p
on
F .
F
S ~ F/p
(or
for some finitely gene
and some finitely generated congruence
The next result is a theorem of Redei
[122,
Theorem 72]. THEOREM 7.9. presented.
A finitely generated monoid is finitely
77 Proof. then
S
If
S = <s^,...,s
is finitely generated,
is the homomorphic image of the free monoid
Z q ; hence
S - F/p
shows that
p
for an appropriate
F =
p , and Theorem 7.8
is finitely generated.
Note that since a free monoid is cancellative, the proof of Theorem 7.9 can be obtained without appeal to any of the material in Section 5 beyond Corollary 5.4. For semigroups that need not be monoids, Theorem 7.10 gives partial results concerning conditions under which a semigroup ring is Noetherian. THEOREM 7.10.
Let
R
be a ring and let
S
be a semi
group. (1)
If S
and only if (2)
R
is finite, then
R[X;S]
is Noetherian
if
R[X;S] is Noetherian,
then
is Noetherian.
If S
is infinite and
R
is a Noetherian unitary ring; the convere fails, even for
S
finitely generated. Proof.
Since
R
is a homomomorphic image of
the Noetherian property in Moreover, R[X;S] so
if R
In R 2
is Noetherian and
is inherited by S
R .
R—module,
is also Noetherian as a ring.
(2) , we first prove that is unitary.
(R/R )[X;S]
,
is finite, then
is a finitely generated (hence Noetherian)
R[X;S]
that
R[X;S]
R[X;S]
R [ X ;S ]
Noetherian implies
Consider the Noetherian ring
.
Since multiplication in this ring is trivial, 2 the additive group of (R/R )[X;S] is Noetherian, hence finitely generated. additive group of
It is also true, however, that the (R/R )[X;S]
is the weak direct sum of
78 2 |S|
copies of the additive group of
that
R/R2 = {0} , so
then follows that converse, R[X;Z R
]
R
R = R2 .
R/R
Since
.
R
This implies
is Noetherian,
has an identity element.
the proof of part
(2)
it
For the
of Theorem 20.7 shows that
is Noetherian if and only if the additive group of
is finitely generated. The final result of this section is in the same vein as
Theorem 7.9 in that it describes all monoid rings over a fixed unitary ring free monoid rings
R
as certain homomorphic images of the
(that is, polynomial rings) over
R .
For
the statement of Theorem 7.11, we introduce the ad hoc terminology pure difference binomial for a polynomial g e R[{Xa >]
of the form
g = X°1...X*£ -
Let
for
•
some nonnegative integers THEOREM 7.11.
xjl...xjn
R
be a unitary ring.
isomorphism, the class o*f monoid rings over
To within
R
can be char
acterized as the class of all residue class rings J/A
A A eA
, where
A
is an ideal of
by pure difference binomials. monoid, then
R [ X ;S ]
ring
If
S
R[{X 1] A
generated
is a finitely generated
is isomorphic to such a residue class
* where
A
is finite and
A
is generated
by a finite set of pure difference binomials. Proof.
Theorem
sults already proved let
S
S .
If
each F .
7.11 is primarily a translation of re into slightly different language.
be a monoid and let { s ^ X e A F
is the free monoid
A , then
S ~ F/p
E™e^Z^
Thus,
a Seneratin S set f°r , where
Z^ - ZQ
for an appropriate congruence
But Theorem 7.2 shows that the kernel
I
of the
for p
on
79 canonical homomorphism of rated by
(Xg -
R [X ;F ] onto
R[X;F/~]
is gene
X^|g ~ h} ; moreover, under the canonical
isomorphism of R[X;F] onto R[{X x >x €a ] » the elements a h X6 — X are precisely the elements of R[X;F] that map to pure difference binomials in each monoid ring over A
R
R[(X^}]
.
This proves that
is of the form
R[{X^}]/A , where
is generated by pure difference binomials.
finitely generated, then which case
A
the congruence
Theorem 7.8.
Part
(5)
can be ~
If
S
is
taken to be finite, in
is finitely
generated
by
of Theorem 7.5 then implies that
is generated by a finite set {X^i -
> whence
I
A
is
also generated by a finite set of pure difference binomials. Conversely, isomorphic to
any such residue class ring
R[X;F]/({Xgot -
X^a I ga »ha € F})
suitable subsets gruence on
F
F *
Let
generated by
Theorem 7.5 shows that and hence
R[{X^}]/A
ring over
R .
R[{X^}]/A for
P
t^16 con"
• Part
R[X;F]/({Xga -
is
(5.)
of
Xh a }) * R[X;F/p]
is, to within isomorphism,
,
a monoid
This completes the proof of Theorem 7.11. Section 7 Remarks
We have mentioned in this section that a general semi group ring
R[X;S]
of a monoid ring
can be defined as an appropriate ideal R*[X;S*]
advantages of working with regarded as an element of
over a unitary ring R1
and
R'[X;S’]
the definition of multiplication in can always t,u e S'
.
be expressed in the form
S'
Rf .
are that
X
Two can be
in this case and, in R f [X;S'] t*u
, each
s e Sf
for some
(In the proof in Theorem 7.1 that
(J) preserves
80
multiplication, s { S + S however;
for example, the possibility that
or that
t ^ T + T
has been ignored.
Not to fear,
the proof still works in those cases.)
The term unital extension of
R
has been used in the
paragraph following the proof of Theorem 7.1 without an ex plicit definition. S
of
R
The term means a unitary extension ring
such that
element of
S .
S = R + Ze , where
The papers
e
is the identity
[25] and [11] contain information
concerning the possible unital extensions of a given ring even in the case where
R
is already unitary.
Under what conditions is
R [ X ;S ]
conjecture would seem to be that only if
R
is unitary and
ditions on since that
R
R
and
S
S
unitary?
R [ X ;S ]
imply that
R [ X ;S ]
unitary implies
R
A reasonable
is unitary if and
is a monoid.
is a homomorphic image of
R[X;S]
Indeed, the con is unitary, and
R [ X ;S ] , it is also true
is unitary.
If
tains a cancellative element, it is also true that unitary implies that For example, let
S
R ,
S
con R [ X ;S ]
is a monoid, but not in general.
S = {a,b,c}
be the semigroup with the
following Cayley table. +
a
b
c
a
a
c
c
b
c
b
c
c
c
c
c
For any unitary ring
R , the element
X a + X^ -
identity element for
R[X;S]
is not a monoid.
, but
S
XC
more on this topic* see [74]. Theorems 5.10 and 7.8 show that a monoid
S
is
is an For
81 Noetherian if and only if semigroup,
S
is finitely generated.
For a
I do not know if either of these conditions im
plies the other.
Similarly,
I know of no results beyond
Theorems 7.10 and 20. 7 concerning the problem of determining conditions under which a semigroup ring is Noetherian.
82 §8.
Zero Divisors
Two problems are considered in this section. is that free
of determining conditions under nontrivial zero divisors ----
of
an integral Theorem
which
8.1.
f e R[X;S]
R[X;S]
that is,
is
R[X;S]
is
domain. This problem is easily settled
in
In the second problem we take a fixed
element
and we seek necessary and sufficient conditions,
usually in terms of the coefficients of f
The first
should be a zero divisor in
R[X;S]
f , in order that .
The more definitive
results on the second problem require some type of restric tion on
S
or on
R.
Theorem 8.1. R[X;S]
R *
{0}.
Proof.
S
Assume first that
admits a total order ation.
Let
f,g
f = r i « l £ ix S l
R
f^
S1 + tl e
and
R R c R for s t “ s+t
The family
We say
s e S , there exists a
of the additive group of
R =
(2)
be a semigroup.
S— graded if for each
Rg
(1)
S
There is a
A = R[X;S] ---
A g = {rXS | r c R}
for each
s e S . THEOREM 8.4. R , where
S
Assume that
{
geneous element of
f
homogeneous element and where for
If
f e R
R .
A nonzero element g n g = » where each
Rg
S-grading of
is annihilated by a nonzero homo
Proof.
the same
is an
is torsionHiree and cancellative.
is a zero divisor, then
position
R c S S 6u
i * j .
g^
of
R
g^
is a nonzero
and
has a unique decom-
g^
do not belong to
To prove the result, we choose
86 a nonzero element
g e Ann(f)
whose number
n
geneous components is as small as possible. of the theorem amounts to saying that total order on Let
S
f = zm f i=l i positions of f
.
We consider f m &g
composition.
n —
.
have
1
Therefore
Consequently,
f.g = 0
If
for
each
^Rs ^ses
RsRu c R s+U
of
<j> : S
{R SRU
and since
follows that
f
and fg = (f +...+f ..)g = 1 m-1
choice of
This
g , we
completes the proof.
S— grading of R , then
of
t,v e T , then
abelian group by
annihilates
consequences of Theorem 8.4, we note
an
T— grading
if
f^g
fgn = 0 .By
At = £{RS I .
Moreover,
, * we obtain a
By induction it follows that
i , so
jective homomorphism
let
be a
components in its homogeneous d e
, and hence n = 1 .
induces a
fm mg &* 0
g , for
f g = 0 m
Before stating some
T
(u) =
A tAy c A t+y .
sur
onto asemigroup
as follows.
That
I <J>(s)=
<J>
each
is torsion— free
is a zero divisor,
and can
then there
Clearly
87 exists a nonzero
element
g e R[X;S]
suchthat
contained in an equivalence class of Proof. T = S/A
The
induces
A and
Supp(g)
fg =
canonical homomorphism <J>
of
a
Since
T— grading of
R[X;S] .
is
0 .
S
onto T
is tor
sion— free and cancellative, Theorem 8.4 implies that
f
annihilated by a nonzero
But
T— homogeneity of Supp(g)
g
T —homogeneous element
g .
is
is equivalent to the statement that
is contained in an equivalence class of
A .
Inci-
j*
dentally, of
if
f , then
f = 2^=i^i fg = 0
COROLLARY 8.6.
t*ie
implies
T “ homogeneous decomposition
f^g = 0
for each
i .
Assume that S is torsion— free and cann sf = .X 1 is the canonical form of
cellative and that
f e R[X;S]
the nonzero element
.
The following conditions
are equivalent. (1)
f
(2)
There exists a nonzero element
rf^ = 0
is a zero divisor in
for each
Proof.
R[X;S]
. r
eR
such
that
i .
That
(2)
implies
(1)
is clear.
The converse
follows from Corollary 8.5, for that result implies that s there exists r e R\{0} and s e S such that 0 = rX • f = E^.^rf^Xs+si . Since rf^ = 0
for each
(1)
If
isprimary in (2)
If
follows
that
i .
COROLLARY 8.7. the unitary ring
S is cancellative, it
Assume that
A
is a proper ideal of
R .
A[X;S]
is a primary ideal of
R
and
A
is primary in
R [ X ;S ] , then
A
S is cancellative. R
and if
S
is torsion— free
88 and cancellative, then Moreover,
if
P = rad(A)
Proof. primary in
If R
cancellative, t * u .
A[X;S]
A[X;S] since
then
is primary in
, then
R[X;S]
.
P [ X ;S ] = rad(A[X;S])
.
is primary in
A = A[X;S] X s (X1 --
R[X;S]
n R .
Xu ) = 0
If
X s (X1 s and no power of X
is
the assumption that
A[X;S]
in
s e S
for some
- Xu ) e A[X;S]
Thus
, then
is
is not
t,u e S
, X1 -
A[X;S]
A
with
XU J A[X;S]
,
.This contradiction to
is primary shows that
S
is
cancellative. By passage to
R [ X ;S]/A[X;S]
consider the case where is a zero divisor in
in
A = (0) . v J
R[X;S]
(2)
Thus,
, it suffices to if
*
f = E11 -f.XSi i=l l
, then Corollary 8.6 implies
that there exists a nonzero element
r e R
rf^ = 0
is primary in
f^
for each
i
. Since
is nilpotent, and hence
proves
(2) .
(0) f
is also nilpotent.
It is clear that
This
P[X;S]
, and
is prime in
by Corollary 8.2.
In
(1)
torsion— free. istic
R , each
P[X;S] c rad(A[X;S])
the reverse inclusion follows since R[X;S]
such that
p * 0
of (8.7), the semigroup For example, and
G
if
K
S
need not be
is a field.of character
is the cyclic group of order
pR ,
then the group ring K[G] is isomorphic to K[X]/(Xpn - 1) n ~ K[X]/ (X - l)p , a special principal ideal ring. Hence each ideal of
K[G]
is primary, but
The same example shows that tain
rad(A)[X;S]
if
S
G
rad(A[X;S])
is a torsion group. may properly con
is not tors ion— free.
In order to continue our investigation of zero divisors of
R [ X ;S ] , we return to Corollary 8.5 and the notation used
89 there.
Thus,
nonzero and
assume that g
is
fg = 0 , where
T— homogeneneous, with
the congruence on S defined by + for some n e Zand some c e S . decomposition of ^g
= 0
f
f
for each
into i .
a A b If
and
g
T = S/A if
are and
A
na + c = nb + c n f. is the i= l i
f = E
T— homogeneous components, then
Let
4>*:R[X;S] -- ► R [X ;T ]
be the
homomorphism induced by the canonical mapping Then
<J>*
R[X;S]
maps each
onto
T— homogeneous element
<J>:S — *T a• Eh.X of
.
m
a monomial
(Eh )X ofR[X;T] . It is i s easy to show, however, that a monomial rX is a zero di visor if and only if either s
is not cancellative in
r S .
is a zero divisor in Since
semigroup, we are able to conclude: ag = ’EgjX ^ , then (Zfij ) (Egj ) = 0 that
Ef^j
where
y
and
Egj
are
y(f^)
is a cancellative s .. if f .= Ef..X and l ii J for each i. We note y (g) , respectively, R[X;S]
integral domain, it follows that either y(f^) = 0
for each
i .
or
T
and
is the augmentation map on
R
.
If
y(g) = 0
In the case where
S
R
is an
or
is cancel
lative, we prove a form of the converse in Theorem 8.9.
The
proof of (8.9) uses an auxiliary result. THEOREM If then
x e R x
8.8.
Let N
is such that is a zero
be the nilradical x
isa zero
of the ring
divisor modulo
R . N ,
divisor.
Proof. Choose y e R\N such that xy e N , say k k k x y = 0 . If xy = 0 , then x is a zero divisor. And if k i k xy * 0 , we choose i maximal so that x y * 0 . Since i k x • x y= 0 , it follows that x is a zero divisor. THEOREM 8.9.
Assume that
D
is an integral domain and
90 S
is cancellative.
such that
S/ a
Let
A
be the smallest congruence on
is torsion— free and cancellative and let
be the kernel ideal of the congruence of zero divisors in Proof. is unitary. onto
S/A
D[X;S/A]
D[X;S]
.
Then
I
I
consists
.
We prove theresult first in the case where
D
Denote by
S
and by
Since
D[X;S]
A .
S
Let
<J>*
<J>the canonical homomorphism of
the induced homomorphism on
is an integral domain, f e I .
I
The proof of part
D[X;S]
.
is prime in (2)
of Theorem m
7.2 shows that each n^
is expresible in the form s• t• is of the form r^(X 1 - X -1) with
f^
f
be the smallest positive integer such that
we assume that
n.
>1
is a zero divisor, of
X
s
-
X
for each
» where s^ A t^.
Let
n ^s i ~ ^ ± ^ 1
i .Before proving that
» f
we consider certain multiples in D[X;S]
t
= h , where s A t . If k is any positive ks kt k-1 (k-i -l)s+jt integer, then X - X = hg , where g =
We note that(k-j-l)s + jt A (k s A t . char D
Hence
=0
—
<J>*(g) = k X ^
or if
char D
and
=p * 0
in either of these two cases.
l)s
and
for each
j
<J>*(g) * 0 p f k ;
since
if thus, g ^
I
We return to the proof that
f
is a zero divisor, considering separately the cases where the characteristic of If 1
char D
D
is, or is not, zero.
= 0 , then the proof above shows that for
< i < m , there exists
f.g. = r. (Xn ^ * ii i g^...gm * 0 If
X
e D[X;S] \ I
= 0 .
Therefore
since this product, is not in
ch ar D
= p
n^ = m^p
such that fg . ..g = 0 l m
n.
, where
.
If n.
e^ > 0
is divisible by and
m^
and
I .
* 0 , then there are two cases
sider, depending upon write
g^
to con p , we
is relatively
91 prime to
p .
There exists
gj I I
such that
figi =
m isi m iti pe i r^ (X — X ) , a nilpotent element since (^igi) nei n is i n i^ i r? (X - X ) = 0 . If n^ is not divisible by p , then there exists is prime,
g. I I 6i T
f.g. = 0 . i6i g k I
it follows that there exists
fg = E^f^g
is nilpotent.
the nilradical of a zero divisor. D
such that
Hence
D[X;S]
f
Since
I
such that
is a zero divisor modulo
, and Theorem 8.8 shows that
f
is
This completes the proof in the case where
is unitary. If
D
is not unitary, let
the quotient field of element then
e .
If
I*
generated by
be the subring of
D
and its identity
is the kernel ideal of
I = I* n D[X;S]
g e D*[X;S]\{0} dg
D
D* = D[e]
.
Thus,
such that
if
on
D*[X;S] ,
f e I , there exists
fg = 0 .
is a nonzero element of
A
D[X;S]
If
d e D\{0> , then
that annihilates
f .
The proof of Theorem 8.9 abounds with examples of zero divisors in
D[X;S]
that need not be in
has characteristic
that is not in subsets of
S
I .
and
In fact, if
D
is not torsion— free, then k-1 (k-j-l)s+jt contains a zero divisor of the form 2j=ox
D [ X ;S ]
0
I .
S
Theorem 8.9 enables us to determine what
can be realized in the form
some zero divisor
f
of
D[X;S]
Supp(f)
for
, at least in the case where
Id| >2 . THEOREM 8.10. with
|D| > 2
that
A =
Assume that
D
is an integral domain
, that S is a cancellative semigroup, and n is an n—element subset of S . Then A
is the support of a zero divisor of
D[X;S]
each
with
a^
is
A— related to some
aj
if and only if j * i .
92 Proof.
The condition given is necessary, for suppose m , where f is a zero divisor. If f = ^ = 1
A = Supp(f) is the
T—homogeneous decomposition of
then Corollary 8.5 implies that each in
D[X;S]
.
f , where f^
T = S/A ,
is a zero divisor
Since a nonzero monomial is not a zero divisor,
it follows that each
a^ e A
is A — related to some
aj
with
j * i . To prove the converse, Theorem 8.10 shows that it is sufficient to prove the following statement: B = {b
...,b^}
is a
k— element subset of
and any two elements of support of an element g =
E^=i(dXb2i -
k = 2r + 1 |D | > 2
of
c + d * 0 b 2r+l
I .
If
where
cX l T
and let
k = 2r
G = (0,a,b,c}.
A = {0,a,b}
is the
is even, take
d e D\{0} .
g = Z. , (dX
|D | > 2
If
D
S
dX
D[X;G]
R[X;S]
is a group.
A—related elements of
+
of order
D[X;G]
G , but with support
2 . S , questions concerning
can sometimes be reduced to the Theorem 8.11 provides two tech
niques for accomplishing such reductions. THEOREM 8.11.
D
is the field of two elements
For a cancellative semigroup zero divisors of
of
is the Klein four-group, then
is a set of
A , is a unit of
-
c,d
is necessary in the state
X a + X b , the only element of
case where
B
k > 1
.
ment of Theorem 8.11, for if
f = X°+
A— related, then
Thus, choose nonzero elements
The assumption that
and
S , where
is odd, this is where the assumption that
such that (cX Lx -
g
are
d X b2i_1) ,
comes in.
b 2r
B
if
Assume that
S
is a cancellative
93 semigroup with quotient group (1)
f
is a zero divisor in
is a zero divisor in (2) of
S
R[X;G]
Assume that
containing
divisor of
S
0 .
R[X;S]
R[X;S]
.
if and only if
f
.
is a monoid and If
f e R[X;S]
G , and let
f e R[X;H]
if and only if
H
is a subgroup
, then
f
f
is a zero
is a zero divisor of
R [X;H] . Proof,
is a unital extension of R , then s is a subring of R*[X;G] and X is a unit of
R[X;G] R*[X;G]
If
R*
for each
se S .
for
b e R*[X;G] .
and
(2) .
Thus,
zero element of XSh e R[X;S] that in
f (1)
Hence
X Sb =
implies
b = 0
We use this fact inproving both
assume that R[X;G]
, and
fh = 0 ,
. There exists
XSh * 0 .
is a zero divisor in
where
h
s e S
(1) is
a no n
such
that
Since
fXSh = 0 , it follows
R[X;S]
.
The other assertion
is patent.
Similarly,
f
a zero divisor in
a zero divisor in
R [ X ;S ] .
nonzero element of
R[X;S]
induces a grading of
R[X;G]
S = G .The ; let
ofb .
T — homogeneous decomposition
of
for each
for some (1) ,
i.
b* e R[X;H]
If
f
b
is
be a
such that fb = 0 ; without loss
T — homogeneous decomposition
fb^ = 0
R[X;H]implies
For the converse, let
of generality, we assume that
in
0
b = Since
0 = fb is
s e SuppCb-^) ,
. Therefore
group
T =
G/H
be the f e R[X;H] n ^1 = 1 ^ ^ . then
b^
0 = fb^ = fX b*
, the Hence = XSb*
, and as
fb* = 0 .
The proof of Theorem 8.11 establishes the following corollary.
94 COROLLARY 8.12. monoid and
H
f e R[X;H]
Assume that
is a subgroup of
S S
is a cancellative containing
, then the annihilator of
f
0 . If
in R[X;S]
rated as an ideal by the annihilator of
f
in
is
R[X;H]
gene .
As an application of Corollary 8.12, we determine cons ditions under which 1 - X is a zero divisor, as well as the annihilator of this element. THEOREM 8.13.
Assume that
is a cancellative monoid.
R
is unitary and s f = 1 - X is
The element
S a
zero divisor in R[X;S] if and only if ns = 0 for some + n e z . If n is minimal so that ns = 0 , then g = 1 +
X S+ ...+ X Proof.
generates the annihilator If f
shows that each divisor. 0 A s —
Since
of
is a zero divisor, then Corollary
A—homogeneous component of 1
that is,
f .
is not a zero divisor, ns = 0
f
it
is a zero follows that
n e z+ .
for some
8.5
On the other
hand, the proof of Theorem 8.1 (or the next paragraph) shows that
f If
is a zero divisor if n
s
is minimal so that
subgroup of
S
of order
and
0
are
ns = 0 , then
n , and
H
A—related. <s> = H
contains
0 .
Corollary 8.12, it therefore suffices to prove that rates the annihilator of fg = 0
is clear.
And if
f
in the case where (1 -
is a
By g
gene
S = H .
That
X S) (aQ + a-|XS+ . . .+an _ ^ n 1^S)=
0 , then a calculation of the coefficients in the product leads to
the system of equations
. . . = a^ , n-l is Ea^X = a^g
sl
0
n- Z
.
Therefore
0 =aQ -
a n _i ~ a i ~
a0
a n = a, = ... = a„ and 0 1 n -1
is in the ideal generated by
g .
=
95 COROLLARY 8.14. R
If
is a unitary ring,
for each
r
G =
is a finite group and
kg* h= E-^X51
then
annihilates 1 -
g
X
g G G .
Proof.
Let m
n .
If
of
(g) in
is a complete set of coset representatives k a- n-1 ig G , then h = (E^= ^X ) (E^ = 1X ) . Thus, Theorem
8.13 shows that
f = 1 —
Xg
andassume that
h annihilates
g
has order
f .
Corollary 8.14 yields an extension of Theorem 8.9 to the case of a coefficient ring with zero divisors. THEOREM 8.15.
Assume
cancellative semigroup. such that
is
a zero divisor of R[X;S] Let
ls-1^
for each
of
for
S .
^i^i=l
^ri^i
extension of If g^ = t^ Hof
* - X
R s^
G .
e
Let
of
R •
and let G ,
G
then
H
=
khf and ^et
annihilates each r e R\(0)
f = E^ r ^(X
each subset
generates a finitesubgroup
1 k
is a ring and S is a ,,n u is a subset of
i , then
R* be a unital
be the quotient group {g*}?
R
-1n
r
If
S
Proof.
s^ A t^
that
^
= S1X ‘ Corollary 8.14 shows that g* 1 — X 1 , and hence hf = 0 as well.
and if
nonzero annihilator
s = E^s^ , it follows that of f
in
rXSh
h If
is a
R[X;S]
.
Section 8 Remarks The reader has probably noted the crucial role played by the assumption that
S is
cancellative in this section.
fact, Corollary 8.5
is essentially the only result about
In
zero divisors in the section
that does not use the cancella
tive hypothesis on
topic of zero divisors of
S .
The
96 R [ X ;S ] , in the case where
S
is not cancellative,
is
largely unexplored territory. Theorems 8.9 and 8.10 stem from the doctoral disserta tion of Janeway
[81].
J an eway’s thesis contains additional
results on zero divisors that have not been included here. Another result that we have not included is a form of the Dedekind-Mertens Lemma for semigroup rings.
For poly
nomial rings, the Dedekind-Mertens Lemma states that if f»g € R [- C } ] , then there exists a positive integer k k+1 k such that c(f) c (g) = c(f) c(fg) . This lemma can be used to give an alternate proof of McCoy's Theorem on zero divisors in
R[(X^}]
.
For a torsion^-free cancellative semi
group
S , it's true that if f,g e R[X;S] \{0> and if k k-1 k = |Supp (g) | , then c(f) c(g) = c(f) c(fg) . The proof hasn't been included in Section 8 because of limited appli cability of the result.
For additional information on the
Dedekind-Mertens Lemma and for its proof, the interested reader may consult
[111],
[51, Section 28], or [54].
97 §9.
Nilpotent Elements
In considering nilpotent elements of again two basic problems.
R[X;S]
, there are
One is the global problem of de
termining conditions under which
R[X;S]
is reduced.
This
is closely related to the problem of determining conditions under which
N[X;S]
is the nilradical of
since
N[X;S]
R[X;S]
, it follows that
only if
R[X;S]
is clearly contained in the nilradical of
(R/N)[X;S]
N[X;S]
is the nilradical if and
is reduced.
The local problem is that
of determining,
for a given element
under which
is nilpotent.
f
set for the nilradical of
f = Er^X
si
, conditions
One of our main approaches to
the local problem is to attempt
to determine a generating
R[X;S]
.
Our first result indi
cates three ways in which nilpotent elements of arise.
, for
Recall from Section 4 that elements
R[X;S]
a,b e S
are
said to be asymptotically equivalent if there exists such that
na = nb
for each
p— equivalent for a prime THEOREM 9.1. N
and that
Each
(2)
If
such that
pr
(3) a rX
If
R
a
K € Z+
and
b
for some
are k ^ 0 .
is a ring with nilradical
is a semigroup. element of
N[X;S] is nilpotent.
a,b e s
are
a,b e S
are asymptotically equivalent, then
p— equivalent and if r e R is a b is nilpotent, then rX — rX is nilpotent.
b -
rX
is
nilpotent for each r e R .
Proof.Statement (3)
k k p a = p b
if
Assume that
S
(1)
p
n > K , while
may
, thereis no loss
unitary since
R[X;S]
(1)
is clear.
In proving
(2)
of generality in assuming that is a subring of
R*[X;S]
,
and R
is
where
R*
98 is a unital extension of
R .
In
(2)
k k p a = p b ,then
, if
(Xa - X b )p k e pR[X;S] , and hence [r[Xa - X b )]p k c p r pkR[X;S] , a a b nilpotent ideal. Therefore r(X - X ) is nilpotent. To prove (3)
, we show that
X a - Xb is
such that
na = nb
for each
show that
(Xa - X^)m
integers such that v £ K , and hence ua
n
> K .
= 0 . If
be
u
If
m = 2K + 1 ,
we
and
v are nonnegative
u+ v = m ,then either ua = ub
+ vb = (u + v)a = ma . (Xa -
K e Z
nilpotent. Let
or
vb = va .
u >
K or
In either case,
Therefore
X b )m = E ; . 0( -l) 1( ? ) X (m'l)a+ib =
x mai™=0( - D x (” ) = x ma(i -
l)m = 0 .
Several necessary conditions in order that
R[X;S]
should be reduced follow from Theorem 9.1, but rather than state them here, we allow them to appear in various theorems as conditions that are both necessary and sufficient in order that
R[X;S]
should be reduced.
There is a natural dichotomy
into the cases of nonzero characteristic and characteristic 0
in considering the nilradical of
the case where char R THEOREM 9.2. acteristic
p
that is,
R
Assume that
R[X;S]
.
We begin with
* 0 . R
and with nilradical
nilradical of
R[X;S]
if and only
is a nil ring, or
(2)
is a ring of prime N.
Then
N[X;S]
if either (1) R * N
and
char is the
R = N — S
is
p— torsion free. Proof. that
(1)
radical of
In view of part
or
(2)
R[X;S]
(2)
is satisfied if .
of Theorem 9.1, it N[X;S]
For the converse,
follows
is the nil-
it suffices, by
99 passage to
(R/N)[X;S]
, to prove that
R[X;S]
is reduced if
R
is reduced and S is p— tors ion— free. Thus, let sA f = If^X be the canonical form of the nonzero element
For any positive integer Pk d V Ef^ XF 1 since
s^
Therefore R [ X ;S ]
; and •pk f^
s-
are not
p— equivalent for
* 0 for each
k , and
9.3.
is
IfI>
each
i
i * j
.
is and
this implies that
p * 0 ,
is a nonzero integral domain of then
D[X;S] is reduced if and only
p— torsion— free.
As in Section 4, we use
g
9.4 to denote the congruence of group
nk f^
is reduced.
characteristic S
Pk f^ * 0 for
this is true since
COROLLARY
if
k , the canonical form of
f .
in the statement of Theorem p— equivalence on the semi
S . THEOREM 9.4.
acteristic
pn .
Let
R
be a ring of prime-power char
The nilradical of
N [ X ;S ] + I , where
N
is the nilradical of
the kernel ideal of the congruence Proof.
R[X;S]
Theorem 9.1
is the ideal R
and
I
is
~ .
shows that
N[X;S]
is con
tained in the nilradical of
R[X;S]
is clear if
R * N , we establish the reverse
R = N , and if
inclusion by showing that
.
+ I
The reverse inclusion
R[X;S]/(N[X;S] + 1 )
is reduced.
By Corollary 7.3, this residue class ring is isomorphic to (R/N)[X;S/p] and hence
.
R/N
Since
R
has characteristic
has characteristic
p .
pn , pR c n
Because
S/g
is
p— torsion— free by Theorem 4.5, it follows from Theorem 9.2 that
(R/N)[X;S/p]
is reduced.
This completes the proof.
,
100 If if
G
is an abelian group of finite exponent
el ek n = p^ •••Pfc
is well
the prime decomposition of
known that
G
decomposes as a direct r . e• . G- - Ig e G | Pj^g = 0^ •
G * G ^ Q - . ^ G ^ , where eA G^ has exponent p^ ponent of
G .
If
n
n , then it sum The group
and is called thep^— primary
G
and
com
is, in fact, a ring considered as a
group under addition, then each
G^
is an ideal of
G .
We
use these facts in the statement of Theorem 9.5. THEOREM 9.5.
Assume that R is a ring of nonzero charei ev n = Pi ••-P^ > and let R = R^S.-.eR^ be the
acteristic
decomposition of R e, char R i * Pi • If
into primary components, where N
is the nilradical of
N[X;S] + £^= 1 ({rXa “ nilradical of Proof. composition radical of
R , then
I r e R^
and
a ~b})
is the
R [ X ;S ] . The decomposition R[X;S] = ^© R^ fXjS]
R[X;S]
the components
R = of
induces the deR[X;S]
, and the nil-
is the direct sum of the nilradicals of
R^[X;S]
.
If
is the nilradical of
R^ ,
then Theorem 9.4 shows that N^[X;S] + ({rXa — of
rX^ | r e R^ , a ~
R± [X; S] ; here
(
)
b})
is the nilradical
is used for ’’ideal of
R± [X;S]
generated by the given set11, but this is the same as the ideal of ical of
R[X;S] R [ X ;S ]
generated by the set.
Thus,
the nilrad
is
(N.[X;S] + ({rXa - rX^ I r e R. and a ~ b})) = i= l i , , 1 Pi N[X;S] + E* , C(rXa - rX | r £ R. and a ~ b}) . 1 1 Pi COROLLARY 9.6. Theorem 9.5.
Then
Let the notation and hypothesis be as in N[X;S]
is the nilradical of
R[X;S]
if
101 and only if
S
is
p^— torsion free for each
is not a nil ring; is reduced and
S
R[X;S]
is
i
such that
is reduced if and only if
p^— tors ion— free for each
R
i .
The results leading up to Theorem 9.5 provide an effec tive device for determining whether a given nilpotent,
in the case where
is as follows.
charR
The decomposition
respect to the decomposition
f^
is nilpotent.
case where
R
then
f
ajXSj
f -
such that
g
f = E^^ajX^
Sj
for
~ Sj
for
f - g -
(a^X
s^
and only if
f
j
the sum
f
First,
of all If
g = f ,
is nilpotent if and
.
We examine
>
s^ .
1 , then >1, then
-
g
.
Assume that the notation is
j sl
with
can
pe
is nilpotent.
Otherwise,
is nilpotent. n s-; g = Sj=i ajX J
f -
If
of
it suffices to consider the
such that a^
p— equivalent tono potent.
f =
is nilpotent if and only if
Thus,
is nilpotent.
only if
f
is
The procedure
has prime power characteristic
we can subtract from monomials
= n * 0 .
R[X;S] = E^_^@R^[X;S]
be effectively determined; each
f e R[X;S]
si a^X
f
f
If it is
is not nil-
is nilpotent if
is nilpotent, and
the latter element has its support of smaller cardinality than
f — g
does.
Continuing this process, we either settle
the question of nilpotency of (n - 1)sjt_
f - g
a composition
Alternately, everything
Supp(g^)
i , while elements of
to elements of
can be
bXs
to
handled
stage, as follows. The congruence g induces u f — g = ^i*iSi °f f ~ g into components
such that all elements of each
before we reach the
step or, at this step we reach a monomial
test for nilpotency. at the
f
Supp(gj)
for
are
Supp(g^) j * i .
p-equivalent for
are not
p— equivalent
102 the
T— homogeneous decomposition of
8, where R[X;S]
T = S/~ .)
If
y
f
considered in Section
denotes the augmentation map on
, then an examination of the proofs of Theorem 9.2 and
9.4 shows that for each
f
is nilpotent if and only if
uCg^) e N
i .
We turn to a consideration of nilpotent elements of R[X;S]
in the case where
ch a r R
= 0 .
be reduced to
the unital case,
tension of
, then the nilradical of
R
contraction to
R[X;S]
f = Ei=ia^XSi , then
Most questions can
for if R*
is a unital
R[X;S]
of the nilradical of f e R Q [X;S]
, where
is
integers.
what follows, we need to use some of
of
Z[ai,...,an ]
FMR— ring.
.
If
Rq - Z[a^,...,an ]
generated ring extension of the ring
perties of such rings
the
R*[X;S]
is a finitely In
ex
Z
of
the pro
Z[a^,...,an ] . The required properties
follow from the fact that it is a Hilbert
We therefore interrupt our treatment of nilpotent
elements in order to present a basic theory of such rings. We stick with the statement made in the preface to the effect that the reader is expected to be familiar with the results of [83].
Thus, Hilbert rings are treated in Section 1— 3 and
elsewhere in [83] and we do not repeat proofs of results that appear there. To recall the definition, ring
R
a Hilbert ring is a unitary
such that each proper prime ideal of
intersection of maximal ideals of
R .
R
is an
It is clear from this
definition that the class of Hilbert rings is closed under taking homomorphic images.
One of the basic results in the
theory is that the class is also closed under (finite) poly nomial ring extension; hence, each finitely generated ring
103 extension
of a
Hilbert ring is again a Hilbert ring,
this applies to Hilbert ring.
the rings
Z[a^,...,an ]
since Z
and
is
a
The other properties of the class of Hilbert
rings we need are contained in following theorem.
THEOREM 9.7. a
The following conditions are equivalent in
unitary ring R . (1)
R
(2)
R [X ]
(3)
For each maximal ideal
traction
is a Hilbert
ring.
is a Hilbert ring.
M n R
of
M
The unitary ring
to R
R
M
M
R[X]
is said to be an
of
, the con
is maximal in
finite maximal residue class rings) each maximal ideal
of
R.
if
FMR— ring (FMR
R/M
The rings
R . for
is finite for
Z[a^,...,a ]
have
this property by the next result. THEOREM 9.8. Then each finitely also a Hilbert Proof. n
=1 .
R[Y]
Assume that
By induction,
it suffices to
= M n R .
finite, and
of
R
is
R[y^]
resolve the case
isa homomorphic
and since homomorphic images of Hilbert
Let
FMR— ring.
FMR— ring.
Moreover, since
FMR—ring. q
is a Hilbert
generated extension R[y^,...y ]
these two properties,
M
R
M
it suffices to show that
be a maximal ideal of
Then M / M q [Y]
M
q
is maximal in is maximal in
Since each maximal ideal of
F[Y]
image
of
FMR— rings have R[Y]
is an
R[Y] and let
R , F = R/Mq
is
R[Y]/M q [Y] ~ F[Y]
.
is generated by an irre
ducible polynomial, the associated residue field is finite.
104 Hence
R[Y]/M
is finite, as we wished to show.
With Theorems
9.7 and 9.8 in
question of nilpotents of THEOREM 9.9.
tow, we return
R[X;S] .
If Sis tors ion—free, then
the nilradical of
to the
R[X;S]
, where N
N[X;S]
is
is the nilradical
of
R . Proof.
If
R
has nonzero characteristic, the result
follows from Corollary 9.6. R*
Assume that
char R
= 0 .
If
is the ring obtained by canonically adjoining an identity
element to
R , then
N
is the nilradical of
R* , so it
suffices to prove (9.9) under the assumption that R is n Si unitary. Assume that f = E^=1f^X e R[X;S ] is nilpotent. f e RQ [X;S]
Then
FMR— ring.
, where
The nilradical
maximal ideals since nM^
.
hence
RQ = Z [f ^ ,...,f ]
For each
Nq
RQ
of
Rq
is
f 6 M^[X;S]
say
Nq =
has nonzero characteristic, and
(Rq/M-^)[X;S] ~ R q [ X ;S]/M^ [X;S]
Therefore
an intersection of
is a Hilbert ring —
X , Rq/M^
is a Hilbert
for each
is a reduced ring.
X , and
f € n(Mx [X;S]) = N Q [X;S] c N[X;S]
consequently,
.
The following ancillary result is used in the proof of Theorem 9.11. THEOREM 9.10. characteristic ideals of
D
0
Assume that and that
such that
D
is an integral domain of
^Px^XeA
nP^ = (0) . A
finite set of prime integers and let of D/P^
A
consisting of all
X
is distinct from each
a Let Aq
prime 1 X— X
be a
be the subset
such that the characteristic of p^ .
Then
"X c A q ^X = (0) •
105 Proof. each
Let
a
e nA€A ^A * Then p,p0 *-*p a € P for 0 1 Z n A A e A , and hence p^...p a = 0 . This implies that
a = 0 , for if not, the characteristic of divisor of
would be a
PiP2 ***Pn *
THEOREM 9.11. characteristic S
D
Assume that
0 .
Then
D
D[X;S]
is an integral domain of is reduced if and only if
is free of asymptotic torsion. Proof.
torsion if
Theorem 9.1 shows that D[X;S]
is reduced.
is free of asymptotic torsion.
S
is free of asymptotic
Conversely, assume that S n si If f = £^= 1 f^X » then as
in the proof of Theorem 9.9, f e D q [X;S] FMR— ring
of characteristic
D q
0 .
for a Hilbert
Thus, to prove that
f = 0 , there is no less of generality in assuming that is a Hilbert
FMR— ring.
distinct elements of
We observe that if
S , then
s ~ t
many primes
p
such that
s^ ~ s^
the subset of
A
acteristic of
D/M^
D .
s
and
for some
t
i,j
{pt}^=1
(0) = nXeAMA » w ^ere
of maximal ideals of
t
are
are not
Thus, there are only finitely
1 ^ i < j < n ; we label this set as a Hilbert ring,
and
for at most one prime
p ; this follows from Theorem 4.6 since asymptotically equivalent.
s
D
.
with Since
CMx^AeA
D
family
Theorem 9.10 shows that if
consisting of all
A
is
A
0 such that the char
is distinct from each
is
p t , then
CO) = n-v a M. . To complete the proof, we show that A 0 f e M^[X;S] for each A £ A q ; this is sufficient since n
M [X;S] = (0) . AgAq A
and that
D/M
y:D — >■ D/M
Thus, assume that
has characteristic
M
m
q ^ ^Pt^l
be the canonical homomorphism,
is maximal in *
D
Let
and let
y*
be
106 the canonical extension of Qk nilpotent, [v*(f)]4 By choice of
q
f e M[X;S]
i .
to
D[X;S]
.
Since
y*(f)
is
0 for some k e Z+ . m k ( p j , the elements q s^ are
and the set
distinct for each and
y
Therefore
yff^) = 0
as we wished to show.
for each
i
This completes the
proof of Theorem 9.11. COROLLARY 9.12. acteristic
0 .
Assume that
D
The nilradical of
is a domain of char D[X;S]
is the kernel
ideal of the congruence of asymptotic equivalence on A semigroup 2x = x + y = 2y
S
S .
is said to be separative if
implies
x = y
for
x,y e S .
The class of
separative semigroups is well known in semigroup theory.
The
next result shows that separative semigroups are the same as the semigroups that are free of asymptotic torsion. THEOREM 9.13.
Let
S
be a semigroup.
separative if and only if In particular,
S
Then
S
is
is free of asymptotic torsion.
a cancellative semigroup is free of asymptotic
torsion. Proof. x,y e S
If
S
is
not separative, then there exist
such that
2x
= x + y = 2y
and
3x = 2x + y = 2y + y =
3y , and hence x
totically equivalent.
This shows that S
S
x * y . Then and y
are asymp
is separative
if
is free of asymptotic torsion. Conversely,
if there existx,y € s
x *
y , then choose k > 1
all
n ^ k .
z =
(k —
Then
minimal
2 (k — l)x =
l)x + (k — l)y .
If
with
such that
2 (k - l)y .
x ~ y but nx = ny
for
Let
z = 2 (k — l)x , then
S
is
107 not separative. S
On the other hand,
still fails to be separative,
2(k - l)y = 4 (k — l)x
and
(k - l)y = 4(k - 1)y =
if
for
z * 2(k - l)x , then 2z = 2 (k - l)x +
z + 2 fk - l)x = 3 (k -
l)x +
4 (k - l)x .
It is clear that a cancellative semigroup is separative. COROLLARY 9.14.
If
domain of characteristic
S
is cancellative and
0 , then
D[X;S]
D
is a
is a reduced
ring. In view of Theorem 9.5 and Corollary 9.12, a description of the nilradical of
R[X;S]
has characteristic
is missing only in the case
where
R
0
main.
Theorem 9.16 fills this gap without explicit mention
either of the characteristic of
and is not an integral do
R
or of any hypothesis
concerning absence of zero divisors in
R .
The strategy of
the proof of Theorem 9.16 is easy to describe.
Let
be the set of proper prime ideals of
R
and let
p^ =
char (R/P.) for each A . Let I_ A PX the congruence £ on S , where ~
be
the kernel ideal of
denotes asymptotic
° equivalence on that
S .
It follows from Theorems 9.4 and 9.11
R[X;S]/(P^[X;S] + Ip ^)
Hence, the nilradical of Q(P^[X;S] + Ip^) element
f
•
R[X;S]
is contained in
of this intersection is nilpotent. is that
plies that some fixed power .
X
Theorem 9.16 shows that, conversely, each
needs to show, of course,
P^[X;S]
is a reduced ring for each
fn
What one
f e X nfP^tXjS] + In a P x) of
f
im-
belongs each
Theorem 9.15 treats two special cases of this
problem. THEOREM 9.15.
Assume that
P
is a prime ideal of
R
.
108 such that
R/P
has characteristic
kernel ideal of the congruence m s f = Ei=i£ ix
(1) to
Sj
p Sj
f e P [X;S ]+ I and if
for
i * j , then
s i>s j € Supp(f)
the relation
f
Proof.
Since each
,
(2) onto
Pk
I
be the
.Let
is
.
k
is such that
that are
e P[X;S] + I
notp-equivalent
implies that
maps to E ^ f ^ X ^ ^
*
[s.l
Let
y
(R/P)[X;S]
, which
f
Pk
R[X;S]
that
and
ideal of
By choice of
in(R/P)[X;S]
.
satisfies the hypothesis of
a ~ b })
, the V p ,
(1) , where the ideal
.
kernel k [y(f)]P P
in (1) im.
Let the notation be as in the paragraph
preceding the statement of Theorem 9.15. R [X :S ]
onto
f. e P for 1
is replaced by the zero ideal of R/P . Therefore (1) k nk plies that y(f^ ) = 0 , which means that fp e P[X;S] THEOREM 9.16.
.
be thecanonical homomorphism of R[X;S] k k . Then y(fP ) = [y(f)]P belongs to
y(I) = ({r*Xa - r*X^ | r*e R/P ~
e P[X;S]
isinP[X;S/~]
for i * j , it follows
J
p s^ =
p— equivalent, then
(1): Under the canonical map from f
[s.] 1 i .
S .
s^
f e P[X;S]
If the positive integer
for any
R[X;S/~]
on
and let
1 e R [x;s]-
If
(2)
~ P
p > 0
The nilradical of
is n
Proof.
A eA
{P [X;S] + I x
p
} . A
We have previously observed that the nilradical
is contained in the intersection described. observe that since nilradical of
N[X;S] + 1^
R[X;S]
For the converse,
is contained both in the
and in each
P^[X;S]
+ Ip^ , we can
109 pass to the residue class ring
(R/N)[X;S/~]
out loss of generality we assume that
R
.
Thus, with
is reduced and that
S
is free of asymptotic torsion. We take an element m s , f = = in[X;S] + Ip } . As in the proof of X Theorem 9.11, it follows that the set A of primes p that say
s^ ~ Sj
for some
A =
with
1 < i < j < m
is finite,
(we presently address the case where
empty).
Part
for each
X
(1)
of Theorem 9.15 shows that
such that
f e n^P^[X;S] = (0) we choose
k
p^ ^ A .
and
f
Thus,
k
p s^ = p sj .
p^
e A.
Part
f t f
e P^[X;S] . If t ^ € Q P^[X;S] = f0) .
if
Take a prime
(2) of Theorem 9.15 = sup{pi >i=1
is
f e P^[X;S]
If
A * <J> , then
si ~ sj » with ^ ideal
A
A = <J> , then
is nilpotent.
large enough so that
k
implies
i,j
such
P^
p e A ,
such that
implies that , it then follows
that
This completes the proof.
Theorem 9.16 generalizes the three previous results C9.4),
(9.5), and C9.12) —
the nilradical of
R [ X ;S ]
of this section characterizing in special cases.
The way to
obtain a given special case from C9.16) may not be clear, however.
It is of some help to have alternate descriptions
of the intersection follows from part and for
P(P->[X;S] + X X (1)
i > 0 , let
Aj * {X e A | p x = p A ) .
In ). One such description Px
of Theorem 7.4. p^
be the
If
Cp^ =
Thus,
ithprime.
let
Define
> then Part
C7.4) shows that
" x ^ ^ s ] and hence
+
p^ = 0
= cP i [X:S] + TPi
’
C1)
of
110 V
a
CPa ^
s
] ♦ I
) A
CP,[X;S]
+ I
) = Px
oo n• i=oCcP i [X;S]
+ ip .) .
A third description of the nilradical — (9.12) as a consequence of (9.16) — nation the proof of (9.16).
the one that yields
follows from an exami
To wit, if
defined in (9.16) and if
f
and
A
are as
| p^ | Al , then the
proof of (9.16), together with
(1) of Theorem 7.4, show that
f e (Ba [X;S] + IQ) n {^n^(Cp[X;S]
+ Ip ) } .
It is clear that
each such set is contained in the nilradical, and hence a third description of the nilradical is Z{(B.[X;S] A
+ I ) n [ n (C [X;S] u pcA P
+I_)]} P
where the sum is taken over all finite subsets
A
of the set
of positive primes. The description of the nilradical in (9.16) is strong enough to enable us to determine conditions under which R [X ;S ] of
R[X;S]
of new on the R
is reduced, and under which .
R
if nx = 0
n
implies
is said to be x = 0
for
is unitary, this amounts to saying that
element of
isthe nilradical
To state these results, we introduce
terminology. The integer ring
N [ X ;S ]
n
R , and in general it means that
THEOREM 9.17. and only if
(1)
totic torsion, and
The R
(3)
p
(2)
S
is regular on
e R ; if
is not element of
semigroup ring R[X;S]
is reduced,
regular
is a regular n
divisible by the additive order of a nonzero
x
one item
is R
R .
is reduced if free of asymp for each prime
m p
such that
S
Proof.
R[X;S]
conditions
(1)— (3)
p such that
empty,
then
R[X;S]
S
shows that conditions
is reduced.
the characteristic
square— free integer.
c
Let
R
element
of characteristic
show that
that
R
duced,
R* .
R
Let
each
V
generated by
is not
p
p
R , and hence
W .
To show
0
is re
or
a ex
= (0) .
W
is
S
is
of
R
In order that
p
R[X;S]
(0)
since p^
p^— torsion— free —
N[X;S]
Thus,
for each
Ip^ =
I = (0) PX p^ * 0 , then
N
is re
such
for the nilradical of
then
Let
R[X;S]
In this case we examine the
This completes the
COROLLARY 9.18.
that
R.y[X;S]is reduced.
isa unit
Therefore, the nilradical of
R .
R
be the regular multiplicative
n(P^[X;S] + 1^ )
= 0 , A asymptotic torsion. If
N[X;S]
Since
be theunital
in Theorem 9.16.We observe that X . If
shows that
is either Ze
is
It is straightforward to
p— torsion— free.
representation R[X;S]
c .
it suffices to show that
S
R
W
Thus, we assume without loss of generality
we further assume that that
of
If
is reduced and that each element of
is unitary.
system in
be the set of
obtained by canonically adjoining an identity
R*
regular in
W
W * .
R* = R ©
tension of e
Let
is not p— torsion— free.
Assume that
are
Conversely, assume that
are satisfied. S
(1)— (3)
is torsion— free and Theorem 9.9
is reduced.
duced,
p— tors ion— free.
Theorem 9.1
satisfied if
primes
is not
is
S
for
is free of
is a nonunit of that is,
Ip = (0) .
n P [X;S] = A
A
proof.
be the nilradical of the ring
should be the nilradical of
R t X ;S ] , the following two conditions are necessary and
112 sufficient.
S
(1)
S
is free of asymptotic torsion.
(2)
p
is regular on
is not
R/N
for each prime
p
such that
p— tors ion— free. Section 9 Remarks
Section 31 of [51] is essentially self-contained,
and it
represents another source for the basic theory of Hilbert rings.
For additional information on separative semigroups
and some reasons they’re studied in semigroup theory, see Section 4.3 of [32] or Section 4 of [74]. The proofs of both Theorem 9.9 and Theorem 9.17 refer to the canonical adjunction of an identity element to a ring
R .
Besides the sources mentioned in Section 7 remarks concerning unital extensions of a ring
R , [51, Section 1] also con
tains information on the adjunction of an identity element to a ring. For noncommutative rings
R , the term semiprime ring
is used instead of reduced ring.
Connell in [36, Theorem 5]
proves the following result.
R
ring and ring n
G
If
is a commutative unitary
is a (possibly nonabelian)
group, then the group
R [G ]
is semiprime if and only if R is semiprime and + is regular in R for each n e Z that is the order of
an element of
G .
Alternate versions of Theorem 9.17, using some of the properties of separative semigroups, may be found in [132], [129], and [30]. Much of the material in Section 9 first appeared in [117].
113 §10.
Idempotents
Given a unitary ring idempotent elements potents —
that
of R
is,
R , the two extremes in terms of are that
Rconsists of
R is a Boolean ring —
are the only idempotents of
R .
or
idem 0
and
1
In the second case,
said to be indecomposable, for in this case
R
R
is
is not ex
pressible as a nontrivial internal direct sum of ideals.
In
Section 17 we examine the problem of determining conditions under which
R[X;S]
is Boolean or, more generally,
in the sense of von Neumann.
regular
We concentrate here on de
termining conditions under which
R[X;S]
is indecomposable.
This is a special case of the problem of determining con ditions under which the idempotents of
R[X;S]
are in
R ,
and this more general problem is essentially no more diffi cult to resolve.
If
f = E^=1fiX Si e R [ X ;S ] , then we
determine several necessary conditions on in order that
f
should be idempotent.
Supp(f)
and
cff)
On the other hand,
Theorem 10.1 is one of the few results that give sufficient conditions for
f
to be idempotent.
THEOREM 10.1.
Assume that
R
is a unitary ring and
S
is an additive semigroup. (1) only if (2) n = |H|
A monomial r
and
s
H
=
If
considered in
(1) (1)
of
R [X ;S ]
is idempotent if and
are idempotent. is a finitesubgroup of
is a unit of
Proof.
group of
rXS
R , then
is obvious.
n
Z^X i
S
and if
is idempotent.
We note, however, that
r
is
as an element of the multiplicative semi
R , rather than the additive group of
R .
In
114 (2) , let
n h. = Z1X 1 .
f
follows that
f2 = nf
hSince X Jf = f
and
for each
(n ^f)^ = n
We remark that each idempotent of and only if each idempotent of
is in
1 - f
contains an idempotent
f ^ R,
if
That
And if
then either
f
or
is anidempotent that is not a monomial, n s• If f = E^-^f^X is idempotent, what conditions must
the coefficients
and the elements s^ of Supp(f) n Some conditions on are easy to establish
satisfy?
(Theorem 10.12).
f^
The main result concerning <s±>
Theorem 10.6, which shows that i .
R
is a monomial.
the first condition implies the second is clear. R[X;S]
it
^f . R[X;S]
R[X;S]
j ,
Supp(f)
is
is a subgroup for each
In this connection, we recall from Section 2 that
is asubgroup of k > 0 .
The
set
S
if and only if s = s + ks
S*
of elements
a group forms a subsemigroup of see this, note that if s + t = s +
S
s = s + k^s
t + k^k2 (s + t) .
statement of Theorem 10.6.
s e S if and
<s>
for some
such that <s>
S*
is
is nonempty; to t = t + k 2t , then
This fact will be used in the
The proof of (10.6) uses two
preliminary results, Theorems 10.2 and 10.4. THEOREM 10.2.
Let
R
be a ring and let
S
be an
additive semigroup. (1) that
Assume that
{s^}^
is a finite subset of
k > 1 is an integer such thatCs^}^ = {ks^}^
for
1 < i < n ,there exists a positive m• such that s. = k 1s . . i l (2) that
Assume that
e = s*L_-|.ri.XSi
R
integer
has prime characteristic
S
and
.
Then
nu < n
p
and
is the canonical form of the idempotent
115
e
of
R[X;S]
.
For each
exist positive integers
i
between
1
and
n , there
s^
= pm is^
m^,k^ < n such that
and
pk i ri = T f Proof. each i
s^
(1):
The equality
= {ks^}^ implies that
is uniquely expressible
between
inductively
1
and
by
s,
n , define
asks^
for some
b^ = i , and
j .
define br + i
= ksu .Since the set (s^)? r r+1 3 1 elements, there exist positive integers h,q with h < q s n + 1 such that s,
= ks, h
b,
=
h
q
= ... = k
b
.
q-h s,
has
n
Then q-h
b _
h +1
Fix
= k
s,
D,
q
h
We conclude that
k q -h s.
= k ^ V ^ s .
= k^s.
bh Moreover,
q - h ^ n
since
{s }n = {ps }!? il il then follow from
and
set
of the multiplicative semigroup of
S
is a
Proof. nu
If
as in Theorem 10.2.
integer
R
k , and for
Therefore each
has prime characteristic
s^ =
k 0
i
To wit, if
, then
are
p
and
in
R .
e = ^ir i^Si Then
is idempotent. Choose km • s^ = p ^-s, for each positive
sufficiently large,
p*cmisi = 0 .
and e £ R .
An alternate elementary proof of (10.2).
R.
p— group, then idempotents of R [ X ;S ] Assume that
.
= {r ^ * . The assertions i 1 l 1 (1) , considering as a sub
(2)
COROLLARY 10.3.
D ps; 1
(r
of
if
.
1
q < n + 1. 2 p n (2) ,we note that e = e = e= Z^ = 1r^X
To prove Therefore
= s,
bh
(10.3)avoids
m € Z+ is such that p " ^
the use
= 0for each
of
THEOREM 10.4. u
e T , and that (1)
If
A
idempotent in u + q
Assume that U
T
is a unitary ring, that
is the subring of
is a nil ideal of
T
T
generated by
such that
T/A , then there exists
u + A
q € A n U
u . is
such that
is idempotent. (2)
Conversely,
idempotent and Proof. = 0 .
Then
q
(1):
if q e T
is such
is nilpotent, then Choose
k
that u
such that
V k T = (u ) © (1 - u)
.
e U
q
+q
is
.
(u - u 2)k = u^(l - u)k
We first determine the
decomposition of 1 with respect to this direct sum. We k have (1 — u) = 1 — b ,where b e (u) . Moreover, 1 - b k k lc divides 1 - b , where b € (u ) . Thus, the decomposition k k k k of 1 is 1 = b + (1 b ) , where b and 1 - b are k k idempotent. We have u = b + (u — b ) , and to complete \r
the proof, we show that u - b is in A n U . k k k k u — b = ( u - u b ) + (ub - b ) . Moreover,
We have
k k k u - ub = u (1 - [1 - (1 - u) ] } = k k k2 u {1 - [1 - k(l - u) + . . . . + ( - 1) (1 - u) ]} , 2
which is clearly in ub^ -
U n ( u - u ) E U n A ; = (u - 1) [1 -
(u - 1) (1 -
also,
(1-- u)^ ]k =
[1 - ku + . . . +
k k (- u) ] } ,
2
which again is in proof of
(1) .
U n ( u - u ) £ U n A .
This completes the
117
(2): u - u
2
We have
= 2uq + q
0 = (u + q) —
2
- q is nilpotent; say 2 k (u — u ) , it follows that
Expanding
k
is an element of the ideal of sequently, the ideal Let
e
I
of
is idempotent,
(u + q) (1
, and hence
2 V (u - u ) = 0 .
U
U
generated by generated by
be an idempotent generator
sider the decomposition u + q
2
7k k k-i 2 i = - zi= l(i)u C“ u )
u
potent.
(u + q)
T = Te @ T(1
it follows that
u ic
k+1
u
.
Con
is
for
idem-
I .
- e) of
T .
(u + q)e
and
Con Since
- e)
ue
are orthogonal idempotents. Weobserve that k k k+1 is a unit of the ring Te since Tu e = Te = Te .
As
qe
is nilpotent,
it follows that
potent unit of the ring element
u(l — e)
1 — e .
Therefore
element of u(l - e) and
unitary ring
onto
Let
Then
A[X;S]
f
A
h + q
annihilates is an idempotent
u + q = (u + q)e
= e
,
is a nil ideal of the
idempotents of R[X;S] (R/A)[X;S]
are
are in
in
R/A
R
if
.
be the canonical homomorphism of .
h e R[X;S]
If idempotents of such that
is a nil ideal of
R[X;S]
Theorem 10.4 implies that there exists that
The
as we wished to prove.
Assume that
(R/A)[X;S]
R , then we take Since
U ,
if idempotents of
Proof. R[X;S]
R .
.
that is also nilpotent; whence
+ q(l - e) = 0 . Therefore is in
ue + qe = e
u
u(l — e) + q(l — e)
COROLLARY 10.5.
and only
Te , and hence
is nilpotent since
T(1 - e)
q =e - u
ue + qeis an idem-
is idempotent.
Thus
R[X;S]
f(h)
is idempotent.
, part
(1)
q e A[X;S]
h + q e R
are in
and
of such
118 f(h) = f(h + q) 6 R/A . For the converse, assume that idempotents of are in
R/A .
clude that part
If
e
is an idempotent of
e = r + a
(2)
for some
R[X;S]
(R/A)[X;S] , we con
r e R , a e A[X;S]
of (10.4), it then follows that
.
By-
a , and hence
belongs to the subring of
R[X;S]
r e R , this implies that
e e R , which completes the proof
of
generated by
Since
(2) . THEOREM 10.6.
is a monoid.
Assume that
Let
S*
be
idempotent of Proof. R[X;S]
.
R[X;S] Let
T = n[r^,...,r ] ring
T
<s>
belongs to
e
as
R[X;S*]
and
an element of
FMR— ring.
1
T
and
then using the fact that (T/M )[X;S]
, where
the other hand,
if
potent because
T
as
Part
(2)
T[X;S]
n .
If
onto
r. i M, 1
for each
e e R [ X ;S * ]
R [ X ;S ]
f^
(T/M^)[X;S] for some
.
X
s^ e S* . r^
On
is nil-
We can therefore write
is nilpotent and
generated by
as asserted.
A
i , then
of Theorem 10.4 then shows that
subring of
X , let
is an idempotent of
is a Hilbert ring. h
R . The t*10
of Theorem 10.2 that r^ e M^
q + h , where
T [ X ;S ] , where
has prime characteristic, we con-
A
(2)
.
, and for each
f^ (e)
T/M.
A
elude from part
Then each
Let
be the canonical homomorphism of between
consisting of
nis the prime subring of
family of maximal ideals of
i
S
S
be a nonzero idempotent of
is a Hilbert
Fix
is unitary and that
is a group.
e =
Consider
R
the submonoid of
all elements <s> such that
e
r .
e ,
g e R[X;S*] e
.
belongs to the
g , and consequently,
,
119 COROLLARY 10.7. group.
If
R[X;S]
Assume that
S
is a cancellative semi
contains a nonzero idempotent
the supporting semigroup of ticular,
S
f
f , then
is a finite group.
In p a r
is a monoid.
Proof.
If
G
is the quotient group of
is a unital extension of zero element of
R*[X;G]
T
and if
R , then considering
f
R*
as a non
, it follows from Theorem 10.6 that
Supp(f) c H , the torsion subgroup of porting semigroup
S
of
f
G .
Thus, the sup
is a finite group.
by definition, we conclude that
Since
0 e S , and hence
S
T e S is a
monoid. COROLLARY 10.8.
If
R
is a ring and
odic semigroup, then idempotents of Moreover, potent of
if
S
Proof.
Let
R[X;S]
is not a monoid, then
R[X;S]
S
0
is an aperi are in
R .
is the only idem-
. T
be a unital extension of
R
and let
S^
be the monoid obtained by adjoining an identity element to 0 S . Since S is also aperiodic, Theorem 10.6 shows that 0 idempotents of T[X;S ] are in T .Therefore idempotents of
R[X;S]
R[X;S]
are in
R[X;S]
n
T .
If
S
n T = R ; in the contrary case,
isa monoid, then R[X;S]
n T = (0)
.
This completes the proof. Under what conditions are idempotents of R ?
in
Corollary 10.8 shows that a sufficient condition for
this to occur is that
S
odic, Theorem 10.9 shows, and
R [ X ;S ]
S
is
aperiodic.
in
the
If
case where
S is not aperi R
is unitary
is a monoid, that the question can be reduced to
120 that in which
S
is a group.
THEOREM 10.9. is a monoid.
Assume that
Let
G
R
is unitary and that
S
be the group of invertible elements of
S . (1)
If the only idempotents
R , then each periodic element of (2) of
Conversely,
if
G
Proof. S .
R[X;G]
To prove
The semigroup
Thus in
1 - X* (1)
are in
Assume that
G
contains
a unique idempotent R[X;S] s
(2) .
R[X;S] If
I = S\G
R[X;S]
. Let
e
be an idempotent of
+ e2 ,where
e^ e R[X;G] and
2
R[X;G]
+
Considered R[X;I] R[X;S]
2
(1
2
6^)62 = (1 - e^ ) e2 = [(1 - e^)e2 ] I
is ideal of and
Then
e^ =
e^ and +
is an idempotent of
2
(1 - ej,)e2 = 0 , and
2
+ e2 = 2e2 + e2 . It follows that
2
e2 = e2 =
write
is aperiodic and contains no identity
element, Corollary 10.8 shows that e2 = 2 e ^ 2
S
- e^)e2 = (1— e^)2e^e2
2
Since
; this
as
e2 c R[X;I] .
(2e-^e2 + e2 ) =e , implying that
e 2 = 2e1e2 + e2 . Therefore
R [ X ;I] .
S .
2
2
hence
is invertible.
2
= e1
(1 —
The hypothesis
is a prime ideal of
groups, this sum is direct, and
2
t .
G = S , the assertion is
abelian
e = e^
.
is in
R [ X ;S ] = R[X;G] + R [ X ;I] .
e
if and only
contains each periodic element of
G * S , then
and we have
R
be a periodic element of
t = 0 ; hence
is sufficient to prove If
are in
s
We show that each idempotent of
clear.
G .
is in
R .
is an idempotent in
implies that
S
R[X;S]
(1) , let
<s>
are those of
contains each periodic element
S , then idempotents of
if idempotents of
of R[X;S]
2
e2^
an idempotent °f
R [ X ;I ] , and again
121 we conclude that
— e2 = 0 .
Therefore
e = e-^ e R[X;G]
The result just established in the proof of
(2)
.
of
(10.9) is of sufficient interest to deserve a separate state ment . COROLLARY 10.10. a monoid.
Let
I * S
no periodic element. R[X;S\I] If
R
is unitary and
be a prime ideal of
S
S
is
that contains
Then each idempotent of
R[X;S]
is in
. G
is a group, Corollary 10.7 implies that the idem-
potents of
R[X;G]
subgroup of of
Assume that
R[X;H]
G .
are in
H
, where
H
is the torsion
Moreover, a fixed idempotent
belongs to
subgroup of
R[X;H]
R[X;K]
generated by
, where Supp(e)
K .
e = E^r^X^i
is the (finite) In Theorem 10.14 we
determine necessary and sufficient conditions in order that each idempotent of
R[X;G]
should be in
R , but first we
establish two results that are used in the proof of (10.14). The first of these, Theorem 10.12,
is of interest in its own
right. THEOREM 10.12. of
R[X;S] (1)
f
is an idempotent element
. The content ideal
(2) unitary,
Assume that
Assume that then either
unitary, then Proof.
R
c(f)
is idempotent.
is indecomposable.
f = 0
or
c(f) = R .
If If
R
R
is
is not
f = 0 . It is clear that
c(f2 ) £ [c(f)]2 .
Since
f
2
isidempotent, we conclude that c(f)
c(f) = [c(f)]
is idempotent and finitely generated,
.
Because
it is principal
122 and is generated by an idempotent element R
is indecomposable.
and
f = 0
or
and
f = 0 .
If
R
Noetherian ring
R
is
Assume that
R .
Assume that
is unitary, then
c(f) = R ; if
THEOREM 10.13.
e .
e
= 0
or
not unitary, then
1
e = 0
A
is a proper ideal of the oo n 00 n = nn=;[A , B2 = n^B^,. . . .
Define
2
There exists a positive integer R
is indecomposable, then Proof.
If
R*
is Noetherian and
k
B^ = (0)
= Q^n...nQt
where
is
A
R
R* .
Then
R , then
of
mary components
Let (0)
in
is the intersection
R, of
C =consisting of those pri
such that |C^|
R*
Hence we assume
is unitary.
B-^
If
k.
be a shortest representation of
the subfamily
then
for some
is an ideal of
P^— primary.
P^ ; assume that
6^ = 6^ .
is a unital extension of
without loss of generality that (0)
such that
A
is not relatively prime to
= k^ , where
B^ = (0)and w e 1re finished.
l ^ k ^ ^ t . If If
k-^
k^ = t ,
< t , then
2
either and
B^ = B^
B? ^
where
and the proof is complete, or else
is the C2 >
intersection of a subfamily C .
Since the set
C
2
of
B2
i=0
that is not in
i. and if
Hence if
H
f e R .
, then ef^ =
is the subring of R
T = n[f1,...,fn ] , then T e C
If f^
generated by and T
is
Noetherian. Moreover, the subgroup is finite, and
H
f e T[X;H] .
of
G
generated by
Each Noetherian ring is a
finite direct sum of indecomposable rings. T = T ]_©... @Tr , where each
n ^Si^x
T^
Thus, write
is nonzero and indecom
posable. With respect to the decomposition T[X;H] = r E^ © T^[X;H] , the idempotent f has a decomposition f = E^f^ , where each f^ e T ^
for each
f^
i , then
is idempotent.
We note that if
f e T £ R .
Also, since each
for
124 T^
is unitary,
|H|
(2) implies that each prime divisior
is a nonunit in each
therefore sufficient (*)
.
To establish
to prove
Assume that
R
To prove H ^©..
(*) ,
is
is a nonzero indecomposable
that each prime divisor of R[X;H]
(1) , it
the following statement (*) .
Noetherian unitary ring and that
the group ring
of
H
|H]
is a finite group such
is a nonunit of
R .
Then
is indecomposable. we express
H
as a direct sum
of cyclic groups of prime-power order, and we use y
induction on where R
p
k .
is
a nonunit of
containing p
of
R[X;H]
potent of
and
H
10.3. m
Thus m
(2)
(j)(f)
<J>
is cyclic of order
Let
M
.
If
4>(f)
f
is a nonzero idem-
is idempotent in
* 0 . The field
R/M
([£) = 1 + M
00 t M ^ = n-^M
indecomposable Assume that
00 t
M2 =
0 .Hence if k =
c(f) = R ,
•
f = 1
for each .
p
by Corollary
m e M[X;H]
is idempotent, me M^fXjH]
m =
(R/M)[X;H] .
hascharacteristic
f = 1 - m , where
€ M2 [X;H ] ,where
,
be a maximal ideal of
of Theorem 10.12 shows that
m e M-jJ X jH] , where
p
be the canonical homomorphism
p— group. Therefore
Whence
clude that
H
, then
H
R .
(R/M)[X;H]
R[X;H]
is a
= 1 — f
and let
onto
Moreover, part and hence
For k = 1 ,
.
Since
t e Z+ .
Similarly,
By Theorem 10.13,. we con , and R[X;H]
is
1 .
R [ X ;H]
is indecomposable for
is the direct sum of
r + 1 • of prime-power order, where
If
cyclic groups H ^ , . . . Hr + ^ . c = P > then we consider
R[X;H] as the group ring of H^ + 1
over
K = H1©...©Hr .
R [ X ;K ]
By assumption,
k = r .
R[X;K]
, where
is indecomposable.
125 Since
K
is finite,
gral extension of implies
p
R[X;K]
R .
is Noetherian and is an inte
Consequently,
is a nonunit of
R[X;K]
yields the desired conclusion that
p
.
a nonunit of
The case
R[X;H]
R
k = 1
then
is indecomposable.
This completes the proof of Theorem 10.14. The analogue of Theorem 10.14 for monoid rings is the following result. THEOREM 10.15.
Assume that
S
is a monoid.
the family of nonzero unitary subrings of that
C
is nonempty.
Let
C
be
R , and assume
The following conditions are equiva
lent . (1)
The idempotents of
R[X;S] are those
of
R .
(2)
The idempotents of
T[X;S] are those
of
T for
each
T e C . (3)
The set
G
of periodic elements of
group of
S ,and if the prime
element of
G
Proof.
, then
p
The equivalence of
(2)
and
is a sub of an
for each (3)
T
e C .
follows from
It is clear that
Cl)
im
(2) , and the reverse implication follows from Theorem
10.12:
if
f c T[X;S]
f
is a nonzero idempotent in
for some
COROLLARY 10.16. R
is the order
is a nonunit of T
Theorems 10.6, 10.9 and 10.14. plies
p
S
R[X;S]
, then
T e C . Assume that
S
is a monoid and that
is a unitary ring of nonzero characteristic
n .
The
following conditions are equivalent. (1)
The idempotents of
C2)
Either
(i)
S
R[X;S]
are those of
is aperiodic or (ii) n = p
R . lr
is a
126 prime power and the set of periodic elements of
S
is a
p-group. Proof.
That
(2)
implies
(1)
follows from Theorems
10.6 and 10.15, for the characteristic of each subring of is a divisor of the characteristic of assume
(1)
ments of
S
Theorem 10.6
of
For the converse,
and assume that the
set G of periodic ele
is nonzero —
is, S is not aperiodic.
shows that
that G
is agroup.
that is theorder of anelement of plies that
R .
p
G
Let
p
be a prime
. Theorem 10.15 im
is a nonunit of each nonzero unitary subring
R , and this implies that
n
is a power of
p , for
has a nonzero unitary subring of characteristic prime—power divisor k , (10.15) then
d >
R[X;S]
1
of
implies that G
For a unitary ring that
R
R
n
.
is a
d
for each
Since
n =pforsome
p— group.
and a monoid
S , it is clear
is indecomposable if and only if
decomposable and the idempotents of
R
R[X;S]
R
is in
are those of
R .
Since the identity element of each nonzero unitary subring of an indecomposable
unitary ring is the same as the identity
element of the ring itself, the next result follows from Theorem 10.15. COROLLARY 10.17. S , the ring
R[X;S]
indecomposable, subgroup of ment of
S
R
and a monoid
is indecomposable if and only if
the set of
S , is a
For a unitary ring
periodic elements of
and the order
of
nonunit of
.
R
S
R
is
is a
each nonzero periodic ele
The characteristic of an idecomposable ring is either
0
127 or a prime power.
In the case of a ring of prime-power char
acteristic, Corollary 10.17 translates to the following statement. COROLLARY 10.18.
Assume that
is an indecomposable v unitary ring of prime-power characteristic p . Let S be a monoid, and denote by S .
Then
R[X;S]
p— subgroup of
S*
R
the set of periodic elements of
is indecomposable if and only if
S*
is a
S .
We conclude the section with a result for group rings that has the familiar ring of (10.14) — (10.18). THEOREM 10.19.
Assume that
unitary ring and that primes Gp
p
G
potent of
R
If
and let
is in
R[X;H]
f e R[X;G]
Gi = Hi @ H 2 , where for each
group ring of f e R[X;H^] R
H^
H2
over
B
be the set of
H = ^peB^p * where G .
Then each idem-
.
is idempotent, then
for some finitely generated subgroup
(H2)p = (0)
Let
p— primary component of
R[X;G]
Proof.
is an indecomposable
is a group.
that are units of
denotes the
R
G-^
of
is a subgroup of
p e B . RfXjH-jJ
f e R[X;G^]
G . H
Considering
Moreover,
and where R [ X ;G ^ ]
, Theorem 10.14 implies that
if each prime integer
q
that is a nonunit of
is also a nonunit of each nonzero unitary subring
R[X;H^] that
qe
.
Let
e
be the identity element of
is a unit of
contents of
e, qe , and
T — t
of Theorem 10.12 shows that
say
as the
T
e = qe • t .
as elements of c(e) = R .
qc(e*t) 5 qR t contrary to the fact that
R[G]
But q
T
of
and assume Computing , part
(2)
c(qe*t) = is a nonunit of
128 R .
This completes the proof. Section 10 Remarks The literature is rich in material concerning idempotents
of group rings ring and
G
R [G ] , where
is nonabelian.
R
is a commutative unitary-
Two general references for some
of this material are Chapter 1 of [121] and Chapters 2 and 4 of [125]. whether
It is unknown in the nonabelian case, for example, R[G]
has only trivial idempotents if
composable and
p
R
is inde
is a nonunit of
R
for each prime
p
that is the order of an element of
G
(cf. Conjecture 2.18
of [125, p. 23]); Formanek in [43] has proved some results related to this question. There is an analogue of (10.15) for semigroups are not monoids.
S
that
It involves use of Theorem 10.6 and intro
duction of the family of subsemigroups of
S
that are
monoids. In general, presentation of the material in this section has followed that of Section 2 of [61]. The description of oo ft nn _^A used in the proof of Theorem 10.13 can be found in Section 7, Chapter IV of [136]. The condition in (10.14) and (10.15) that
p
is a n o n
unit of each nonzero unitary subring of
R
the condition that the ideal
contains no nonzero
idempotent.
pR
of
R
is equivalent to
129 §11.
Units
All rings considered in this section are assumed to be unitary, and all semigroups are assumed to be monoids. area of units of monoid rings is vast — the topic of a monograph in itself —
The
large enough to be
and hence some restric
tions on the scope of treatment here are necessary.
The main
restriction we impose is in considering primarily torsion—free monoids.
In this case, we are interested in characterizing n s, the units u = E^r^X of R[X;S] in terms of the coeffi cients
r^
and the elements
a description of
s^
of
Supp(u)
the Jacobson radical of
case.Another problem of significant
.
We also seek
R[X;S]
in this
interest is that
giving conditions under which each unit of
R[X;S]
of
is
trivial, where by a trivial u n it , we mean one of the form rXs
where, necessarily,
vertible in
S .
r
is a unit of
R
and
s
To begin, we consider the case where
is in S
is
both torsion— free and cancellative. THEOREM 11.1. with identity and
Assume that S
D
is an integral domain
is a nonzero torsion— free cancellative
monoid. (1) then
f
If and
f,g e D[X;S]\(0) g
fg is a monomial,
are monomials.
(2)
D [ X ;S ]
admits only trivial units.
(3)
D[X;S]
is semisimple.
Proof.
(1):
Let
0
and
S
defined
b + c = e + f ,
is torison— free and aperiodic, but not cancella
tive; to within isomorphism,
S/~
can be thought of as the
iilc x yJ z
multiplicative monoid
of terms
R[X,Y,Z]/(XY - XZ)
The next result gives an equivalent
.
in
description of a torsion— free aperiodic monoid. THEOREM 11.7. if and only
if
Proof.
The torsion— free monoid
0
S
is the only idempotent of
Clearly
0
For the converse, assume that
idempotent of
S
The semigroup
<s>
that
and let
s
Since
of 0
S
if
S
S
is the only
be a periodic element of
contains an idempotent
by hypothesis.
S.
is the only idempotent
is aperiodic.
0 = kO
is aperiodic
ks , and
S
.
ks =
is torsion— free, it follows
s = 0 . If
S
contains a nonzero idempotent
11.4 need not extend to the monoid ring example,
1
- 2
is not nilpotent in
.
For
2
and
Z .
Theorems 11.9 through 11.12 is a special case of
Theorem 11.13.
We assume throughout these results that
is a unitary ring, monoid,
R[X;S]
- 2XS e Z[X;S] is a unit of order
s { G , but Each of
s , then Theorem
and that
that G
S
R
is a torsion— free aperiodic
is the set of invertible elements of
S .
For the sake of brevity, we do not repeat these hypotheses in
136 the statements of (11. 9)-( 11.12) .
The proof of (11.9) uses
an auxiliary result that has already been proved in
(2)
of
Theorem 1.1; we repeat the statement of the result in the form in which it is used here. THEOREM 11.8. monoid Then
S I
Assume that
and that
I
is contained in an ideal
p * 0
istic
of
disjoint from S
T .
maximal with
T , and each such
Assume that
and
assume that
proper prime ideal of where
f e R[X;S*] Proof.
exists
S
.
, then
t e S*
J
is a prime
S*
I
t + kt = kt
I
is empty. 1=
{s e
, then
If not,
S |t + s =
is a proper ideal of
of
A
S , it S .
Therefore
[s.], [t] £ T
ns + c = nt + c
n(s + c) = n(t Thus
S
such that the We prove that are such that
for some
+ c) and
[s] = [d]
S
is empty.
is cancellative.
Thus, if
s}
S ,and
k e Z+ , contrary to the fact that
T = S/~
,
A = {s e S | there
be the smallest congruence on
is torsion— free. [x]
such that
We conclude that
is torsion— free.
Therefore
is a unit ofR[X;S]
meets the semigroup
factor semigroup
n[s] = n[t]
1+f
unique proper prime ideal of
for some
is aperiodic. ~
S* = S - {0} is the only
s = s + t}
t e S*
is the
follows that
Let
is a field of character
f = 0 .
such that
is also nonempty.The set since
If
R
We first claim that the set
then there exists
Let
J
S
S . THEOREM 11.9.
T
is a subsemigroup of the
is an ideal of
respect to failure to meet in
T
c
s + c = t + c and
be an invertible element of
T
£ S . since
is tors ion— free.
T , say
S
137 [0]
= [x] + [y] = [x + y]
.
is empty, we conclude that
Then
x + y
x + y = 0 .
~
0 , and since
Because
S*
only
proper prime ideal of
S , it follows that 0
only
invertible element of
S .
[0]
is the only invertible element of
Therefore
is the is the
x =y =
T .
A
0 , and
Summarizing,
T
is a torsion— free cancellative monoid with no invertible ele ment other than
0 .
the only units of
It then follows from Theorem 11.6 that
R[X;T]
are those of
the canonical homomorphism of <J>(1 + f) = 1 + (f) e R • is empty.
Therefore
s^,sj e Supp(f)
R[X;S]
But
R[X;T]
Let
s^ + t .j = Sj + t^. . Let
over
all
<J>
is
, then
since
f = E^r^Xsi .
s^~ s^. , then we choose
that
s^ , s. e
onto
Thus, if
[0] ^ Supp(<J>(f))
<J>CfD = 0 •
and if
R .
A
If
t ^ e T such
t= £t^. > where the sum is taken
Supp(f)such that
s^ ~ s^. . It
then
follows that consider
s. ~ s. if and only if s- + t = s. + t . We 1 J 7 1 J X tf = Z^r^Xsi+t = Z^a^X^ , where bj,...,bm are
distinct and where
b. / b. 1
for
i * j . Then
3
0 = ■ W Because each each
a .
ta
T
T
0 .
An element
if ntc S t e T
containing
in
T .
If
for some
Sq
S
S
be a submonoid of
S
;S Q
S =
is said to be integral
n e
Z+ . The set SQ
S q , we say SQ
S
be integrally closed if
s e S
the case where
t eT such
of all
is integrally closed in T
S = SQ .
S
S
and
S
In defining
quotient group of
S . In
is almost integral over
that
s + nt e S
for each
S
monoid of
T containing S , and
closed in
T if
S = S*
.
completely integrally closed
S = S* .
S , then
S*
S , and If
integral over
T .
T .
In S ,
is said to the concepts
is cancellative and
the complete integral closure of
closure of
S
is the quotient group of
n e Z+ .
t eT S
The set
S*
is called
it is a sub
is completely integrally
If
T
S*
need not be
is the
is called the complete
S
quotient integral
is completely integrally closed if
is integral .
T ;
S
In general, in T .
is a
Sif there exists
rn S
T
this case, an
t e T that are almost integral over
group of
of
to integrality for almost integrality, we r e
submonoid of the element
is a submonoid
is integrally closed in
is called the integral closure of
strict to
S
of
is called the integral closure of
is cancellative and
corresponding
T
t e T
that are integral over
It is easy to see that case
^or
® V nBX^a ■ W BX)T •
be a monoid and let
containing
elements
= ^nABA^ ta
Consequently,
Let
S
■
is regular in T ,
"A = W
over
•
over
S, then
t
The next result establishes the
is almost converse
152 under appropriate hypotheses on THEOREM 12.4. be
a submonoid of (1)
t
group of
Xt e R[T]
If
satisfies
and
T .
be a cancellative monoid, R
be a unitary
is integral
over
let
S
ring.
R[S] if and only
if
S .
Assume that S .
T
T , and let
is integral over (2)
S
Let
S
Tis a submonoid of the
t eT
a.c.c.
quotient
is almost integral over
for ideals, then
t
S
and
if
is integral over
S . (3)
If
R
is an integral domain and
of the quotient group of of
q.f.(R[S])
R[S]
Proof. X*
(l):
Xt
is almost integral over
is almost integral over
We show that
is integral over
let
R[S]
fore
for some
s e S
for some
is cancellative.
S
if
; the converse is clear. Thus,
nt e Supp(h^Xlt)
nt = s + it
S .
t is integral over
Xnt + h 1X (n’1)t+ ...+h n = 0 ,where each n-1 0 7
It follows that
T
t
is a submonoid
S , then considered as an element
, an element
if and only if
T
This proves
and
that
h. £ R[S] l
i < n .
(n - i)t = s t
There since
is integral over
S . (2):
Let
s e S
be such that
+
n e Z
.
The ideal
{s +
un = itCs + nt) + S] of
for some
for some
(3): if
m .
n < m + 1 .
and therefore
R[S]
s + nt e S
for each
°o
t
Hence
the converse.
s + ( m + l ) t £ S + n t + S
It follows that
(m + 1
- n)t £ S ,
is integral over S .
The assertion that t
S is generated by
Xt
is almost integral
is almost integral over Thus,
let
f £ R[S]\C0)
R
is clear. be such that
over
We prove
.
153
fXnt e R[S] that
for each
s + nt e S
integral over
for all
R
element of
s e Suppff)
If
n , and therefore
Assume that
is a subring of T .
Let
U
T
T
t
is a unitary integral d o
be a torsion— free cancellative
S
g e T[U]\(0)
has canonical form
be a submonoid of
U . Assume
g =
and each
g^ u^
that .Then
belongs to the complete integral closure of
R
is almost
containing the identity
monoid and let
only if each
, it follows
S .
THEOREM 12.5. main and
n e Z+ ,
R[S]
g
if and
belongs to the complete integral closure of belongs to the complete integral closure of
S .
Proof.
Denote by
integral closures of g. € R* si
and
si e S each
(R[S])* , R* , and R [S] , R , and
u. e S* 1
for each
are chosen so that k e Z+ , then let Clearly
each
, and hence
k
e Z+
Conversely, fgn e R[S]
order on write
U
if
rXs
g
S .
Assume first that
i .
If
r. e R\C0) l
€ R
and
+
g e (R[S])*
c (R[S])*
for each
.
is
f^X5!
Therefore
This implies that belongs to
for
Let
^
be such
be a total
compatible with the semigroup operation, and
+ k e Z
s =
f e R[S]\{o)
f, * 0, s, < s~ 0 v ,
> E^k^vCs^) = ( ^ k ^ )vfs^) = nvfs^) .
where w e Supp(a^)
v(s) = v(w)
or to
+ h
s e SuppCa^f1)implies
proved that
To o b
s c Supp(fn - X ^ l h ^ )
On the other hand,
quently,
.
•
Supp(fn - X ^ l h ^ )
i < n .
is of j
nvfs^)
, where
s
Supp(a^f^)
Xnslhi * 0 .
v— value
g = ( f n - xnsi h ? ) + Each such
0,
Therefore
tain a contradiction, we show that s e Supp(g)
+ an = 0
v(c)
s Thus
= w + c v(w)
> ivCs^) .
+ v(c) > ivCs^) > n v C s ^ Supp(fn + Z^ ^a^fn
^ 0 , and Conse
. We have
, and hence
This completes the proof.
,
158 The analogues of Theorem 12.5 and Corollary 12.7 follow easily from the two preceding results. THEOREM 12.10. R
is a subring of
Let
U
(1) R
is a unitary domain and
containing the identity element of
T .
S be a
U .
R[S]
is integrally closed in
is integrally closed in T and
S
T[U]
if and only if
is integrally closed
U . (2)
The integral
R ’[Sf] , where S'
R'
closure of
is
R[S]
(1) :
is clear that
If
R[S]
R = R1
Since
U
is R
in
in T
S = S'
and
U .
is integrally closed
and
verse, assume first that Sf =A u r a va
in T[U]
the integral closure of
is the integral closure of S Proof.
it
T
T
be a torsion— free cancellative monoid and let
submonoid of
in
Assume that
.
is a group.
in T[U]
To prove the
,
con
By Theorem 12.8,
is an intersection of valuation monoids on
U .
] is integrally closed in T[U] for each a by a Theorem 12.9, it follows that the integral closure of R [ S ] in
T[U
T[U]
is contained in A T [ U ] = T[nU ] = T [ U ’] . To a va a va complete the proof in the case where U is a group, it therefore suffices to prove that closed in
T [ U f] .
showing that
is integrally
Simplifying the notation,
R[S]
is integrally closed in
integrally closed in f e T[S]
R ’[U1]
T
integral over
.
this amounts to T[S]
if
R
To prove this, take an element
R[S]
.
If
G
is the quotient group
of
S , then there exists a finitely generated subgroup
of
G
such that
f e T[H]
is
and
f
is integral over
As in the proof of Corollary 12.6, however,
H ^ Zn
H
R[H]
.
for some
159 n .
It is known that R
that
R[X^,...Xn ]
hence
R[H]
integrally closed in
is integrally closed in
~ R[Zn ] ~ R[X ^,...»Xn ]g
each coefficient of
f
pletes the proof of
(1)
If
U
. is in
R[S]
closure of
S
R[S]
is the multiplica
Therefore
in
f e R[S]
let
is
W .
in T[U]
f e
W
U
R[H]
R 1[M]
This com
is
a group.
be the quotient groupof
,where
Therefore, is
R'[M]
M
is the in
the integral
n T[U] = R'[M n U] =
R' [S' ] . Statement
(2)
COROLLARY 12.11. and let
S
Cl) D*
D[S]
D
is
D'[S']
, where
and S'is the integral
is integrally closed if and only if
D
and
are integrally closed. Proof. (2)
We need only
of Theorem
gral closure of group of
that
D [S ]
S .
S
D ’[G]
be a unitary integral domain
The integral closure of
is the integral closure of
(2)
(1) .
be a torsion— free cancellative monoid.
closure of
Part
of (12.10) follows easily from Let D
S .
D[S] Since
prove (1) ,
for
12.10shows that
C2)
then follows.
D f[S']
is the inte
inD 1[G] , where
G
is
the quotient
D 1[G] is an overring of
D
and since
is integrally closed by Corollary 12.6, we conclude D'[ S’]
is the integral closure of
,
.
implies that the integral
in T[W]
tegral closure of
B
R , and
we
U .The case just considered closure of
T[X^,...Xn ] , and
in the case where
is not a group,
implies
is integrally closed in
T [ X 1 ,...,Xn ] g ~ T [ Z n ] ~ T [H] , where tive system generated by
T
D[S]
.
Having settled the problems of determining conditions
160 under which a monoid domain
D[S]
is integrally closed or
completely integrally closed, we conclude this section with a description of the integral closure of THEOREM 12.12. cal
N
and let
Assume that R
S
R
f
Then
tegral over
f
is integral over
is nilpotent for each R
s * 0 .
in
R[S]
It is clear that each element of
integral over
R . R .
.
be a torsion— free cancellative monoid.
is the integral closure of
Proof.
in R[S]
be a unitary ring with nilradi
f = ^ses^s^S 6 R [S] •
if and only if
R + N[S]
Let
R
g = f - fQ
.
R + N[S]
For the converse, assume that
Then
Thus,
f
is
is in
is also integral over
R .
Let
(*)
gn + r n- i « n" 1+• • -+r i g + r o =
0
be an equation of integral dependence for
g
over
P
<J>
is the canon
is a proper prime ideal of
ical homomorphism of
R[S]
R , and if
onto
(R/P)[S ] , then
R .
(*)
transformed to an equation of integral dependence for over
(R/P) [S] .
It follows that
take a total order deg 0
or
£
on
S .
ord (g) = 0
If
is the nilradical of
R[S]
it follows
We note in by Theorem
161
Section 12 Remarks For polynomial rings, Theorems 12.5 and 12.10 and Corollaries 12.6 and 12.11 are known of [51]).
(see Sections 10 and 13
But for rings with zero divisors, equivalent con
ditions for the polynomial ring to be integrally closed or completely integrally closed are not known.
If
R [X ]
is
integrally closed, then
R
is integrally closed and reduced,
but the converse fails.
For more on this topic, see [3].
Power series rings over integral domains behave like monoid domains in regard to the properties of complete inte gral closure covered by Theorem 12.5 and Corollary 12.6. But for integral closure, the situation is quite different. For example, for fields
F
and
K
with
K [[X]]
need not be integral over
D [ [X]]
need not be integrally closed if
closed.
Of*course,
closed, then grally closed.
if
D
D , and hence
K/F
F [ [X]] . D
algebraic, Moreover, is integrally
is Noetherian and integrally D [[X]] , are completely inte
Again Section 13 of [51] and its references
provide additional information on this topic.
162 §13.
Monoid Domains as Prufer Domains
The closely related classes of Prufer domains, Bezout domains, Dedekind domains, and principal ideal domains are of fundamental importance in commutative ring theory.
In this
section we determine necessary and sufficient conditions in order that a given monoid domain of the four classes named. Section 13 that that
S
D
D[S]
should belong to one
Thus, we assume throughout
denotes a unitary integral domain and
is a torsion— free cancellative monoid.
We begin
with the problem of determining conditions under which
D[S]
is a Prufer domain. THEOREM 13.1. D
Assume that
and that each ideal of D
is a Prufer domain, then Proof.
is contracted from
the quotient field of
D
is contracted from
D .
J ,it
conclude that
(a,b)
COROLLARY 13.2. If
H
is
Prufer domain.
J
D .Let
K
J n K = D , and
is a Prufer
domain,
D ,we
as we wished to show. D[S]
containing
In particular,
a,b e
Contracting to
2
Assume that S
If
is integrally
for all
Since J
= (a ,b )
a subgroup of
D
follows that
(a,b)2J = [(a,b)J]2 =(a2 ,b2)J . 2
J.
Since each principal ideal of
is integrally closed.
2
domain of
is a Prufer domain.
(a,b)2 = (a2 ,b2)
be
D
is an extension
It suffices to show that
closed and that
hence
D
J
is a Prufer domain. 0 , then
D = D [{0>]
D[H] is
a
is a Prufer
domain. Proof.
Theorem 12.2 shows that each ideal of
contracted from
D[S]
.
D[H]
is
163 THEOREM 13.3.
If
domain, then either Proof. such that
D
s
is not
invertible in
S
i
s % Suppfx
ficient since
f^
S
g)
and if
D [S] S
invertible.
is a group.
d e D\{0)
If
.
S
s
is not
is cancellative, it 2
Therefore
d = d f^
We conclude that
d
D
e s
s
, then
dXs = d 2f + X 2sg . Since
f , and consequently,
1 = df^ .
is a Priifer
is not a group and choose
and since
2s
of
0
is a field or
Assume that
dXs e (d2 ,X2s) , say
that
S *
follows
for some coef is a unit of
is a field if
S
D is
not a group. Recall that a group nx = g
is solvable in
An arbitrary group group
H
phism.
H* Thus
can
is divisible
for each
g e G
If
H
H*
and each
G .
is uniquely determined
Hp , where
is a vector
n e Z+ .
As is implied by up to isomor
istorsion—free, then considering H
Z , each nonzero element of is merely
if the equation
be imbedded in a minimal divisible
H* called the divisible hull of
the terminology, H*
over
G
G
T
is regular on
H , and
is the set of nonzero
integers.
space over
Z
as a module
Q , and as a group is isomor
phic to a direct sum of copies of the additive group
Q .
The statement of Theorem 13.4 involves the divisible
hull.
THEOREM 13.4.
Assume that
group with divisible hull H* .
H
is a nonzero torsion— free
The following conditions are
equivalent. (1)
D[H]
is a Priifer domain.
(2)
D[H*]
(3)
D
is
is a Priifer domain. a field and
H* = Q .
164 (4)
D[H]
Proof. implies
is a Bezout domain.
(1) < = > (2) :
(1) .
For the converse, assume that
Priifer domain. Then
Hm
For
m e Z+ , let
is a subgroup of
H = u00 H m= 1 m D[Hm ]
Corollary 13.2 shows that (2)
, where
H*
D[H]
is
Hm = {g e H*|(m!)g
isomorphic to
H, c H 0 c . . . . 1 “ 2“
H
a
e H) .
and
It follows that each
is a Priifer domain, and hence D[H] = u~D[Hm ]
is also
a Priifer domain. (2) G
of
D[Q]
=>
H* . is
(3) : Then
We write
H*
as
D[H*] a D[0][G]
a Priiferdomain.
.
Since
valent,
D [Z ]
a field,
take a nonzero element
Q © G
for some subgroup
By Corollary 13.2,
(2)
is also a Priifer domain. d e D
and (1)
are equi
To show that and let
D
is
A =
(d2 ,(l - X ) 2) be the ideal of D[X] generated by d2 and ? (1 - X) . Since A and XD[X]are comaximal, it follows that
A
is
contracted from
have
d(l - X) e AD[Z]
f
and
g
unit of
D .
is a Priifer domain.
f,g
Consequently, that
some
Consequently
= A .
Write
t D[X]= D[1 - X ] 1 - X
d e d D , and this
we conclude G^ .
D[Z]
n D[X]
as polynomials in
it follows that
= D[X,X *] = D[Z]
since
Therefore d(l - X) e AD[Z] d 2f + (1 - X ) 2g , where
D [ X ] ^
d(l - X)
implies that
D[Q] .
Therefore
proof that
(2)
1 - X
d
D,
is a
Returning to
G = {0} , for if not, then
is not a field because
=
Regarding
G = Q © G^
H* , for
is a Priifer domain, and
this is a contradiction to what was just proved — D[ Q ]
We
with coefficients in
Dis a field.
D[Q @ Q]
.
,
to wit,
is a nonzero nonunit of
G = {0} , H « 0 , and this completes the implies
(3) .
165 (3) by
=>
(4) :If Gm
1/m! for each
u?G. 1 m
Therefore
m > 0 , then
and
D D[Q]
and
generated and
D[G ] ~ m
D [ X ], a principal
is a field.
of Q
£ G2 E •••
D[Q] = u"d [G ] 1 m
a quotient ring of since
is the subgroup
Therefore each
Q =
DfX.X'1 ]
ideal domain
(PID)
D[G ] m
PID ,
is a
is
is a Bezout domain.
(4)
implies
(1) because a Bezout domain
is a Prufer
domain. If then
G
is any abelian group —
G is a
torsion— free or not —
Z—module and we can consider the group
a vector space over the field
Q =
, where
G^, as
T = Z \{0> .
The dimension of this vector space is called the torsion— free rank
of G
and is denoted by
ro(G)
>
charac-
terized as the supremum of the cardinalities of free (that is, linearly independent) subsets of and only if group
G
is
is a subgroup of
rank of
r o(G)
We use
If
D[S]
of
S , then
r^fM)
that
D is a field and
G
M .
is the quotient group D[S]
, and hence
also a Prufer domain.ByTheorem 13.4, it follows
Moreover, that either
if
rQ (S) S
S £Q q
=1
if
D[S]
is
a Prufer
is not agroup, then Theorem 2.9 or
- S
set of nonnegative rational numbers. then
.
is the quotient group of
is a quotient ring of
is
shows
> where G
Q
the torsion— free
to denote the torsion— free rank of
D[G]
domain.
H
wedefine
is a Prufer domain and if D[G]
^
if and only if semigroup M ,
M .
ro(G) = 0
for a torsion— free
For a cancellative to be
Thus
a torsion group, and
H , rQ (H) = 1
M
G .
£ Q q , where Q q Thus, if
S is isomorphic to a submonoid of
Qq
is the
S * {0}
containing
, 1 .
The next result determines conditions under which the monoid
166 ring of such an
S
THEOREM 13.5. is a submonoid of tient group of
, over a field F ,
is a Prufer domain.
Assume that
a field and that
Qq
S .
Fis
containing 1 .
Let
G
S
be the quo
The following conditions are equivalent.
(1)
S
is integrally closed.
(2)
G n Q0 - S .
(3)
S
is the union of an ascending sequence of cyclic
monoids. (4)
F[S]
is a Bezout
domain.
(5)
F[S]
is a Prufer
domain.
Proof.
(1) = >
over
Zq £ s .
over
S , and
(2) :
Each element of
Hence the elements of G nQ q = S
(2) =“ > (3) :
since
Part
(1)
S
Qq
G n Qq
is integral are integral
is integrally
closed.
of Corollary 2.8 shows that
is the union of a chain of cyclic groups
{ ( g ^ ) •
out loss of generality, we assume that
> 0
Then
S = G n q q = [u"(g-)]
where
- ^ i +l ^ (3) “ > (4) :
" Q0 =
^or each
G
With
for each
i .
n Q01 = ^ i *
’
i •
It follows from
(3)
that
FI S]
is the
union of a chain of subrings, each isomorphic to F [Zq ] = F[X] , a That
(4)
PID . implies
is a Prufer domain, then S
Therefore (5)
F[S]
is clear.
F[S]
is a Bezout domain. Moreover,
if
F[S]
is integrally closed, so that
is integrally closed by Corollary 12.11. We note that condition
trinsic property of
S
that
assumptions that S c Q q submonoid
T
of
Q
and
(3)
of Theorem 13.5 is an in
requires noreference to that
a Prlifer monoid
1e S
. Wecall a
if T
the nonzero
is the union
167 of an ascending sequence of cyclic submonoids.Theorem is a summary of the results of this
section up
the hypotheses of Section 13 concerning
D
13.6
to thispoint;
and
S
are re
peated in its statement. THEOREM 13.6. main and that monoid.
S
Assume that
D
is a unitary integral do
is a nonzero torsion— free cancellative
The following conditions are equivalent.
Cl)
D[S] is a Priifer domain.
C2)
D[S]
C3)
D
is a Bezout domain.
is a field and to within isomorphism,
either a subgroup or a Priifer submonoid of
S
is
Q .
A Dedekind domain can be characterized as a Noetherian Priifer domain, and a
PID
is a Noetherian Bezout domain.
In view of these facts, it is easy to determine from Theorem 13.6 conditions under which
D[S]
aPID
proof of (13.8) uses a basic
CTheorem 13.8).
The
is a Dedekind
domain or
result that is also referred to in Sections 14 and 15. THEOREM 13.7. contains in
If
<s>° , then
D[S] .
Thus,
if
s,t e S (1 - X1*) D[S]
are such that ^ properly contains
satisfies
condition for principal ideals
properly (1 - X s )
the ascending chain
(a.c.c.p.)
, then
S
satis
fies the ascending chain condition on cyclic submonoids. Proof. 1
We have s = nt
- Xs = (1 - X t)(l +
nonunit of
DIS]
properly contains
for some
n > 1 .
X t+...+ X (n‘1)t)
by Theorem 11.1. (1 - X s) .
Therefore ,
Consequently,
where is a (1 - X t)
168 T H E O R E M 13.8. Theorem
13.6.
The
and
(2)
D[S]
or to
is a
D[S]
S
be
as
following conditions
D [S ] is a Euclidean
(4)
in the s t a t e m e n t
of
are e q u i v a l e n t .
domain.
PID .
is a Dedekind domain.
D is a field and
S
is isomorphic either
to
Z
Zq . Proof.
=>
(4)
We need only establish the implications and
domain, then
D
a submonoid of is Noetherian that
D
(1)
(3)
(3)
Let
S
(4)
=>
is a
(1) .
Thus, if D[S]
field by Theorem 13.6.Moreover,
0 , and
S
If
that is,
S
is a group,
S - Z .
group, then Theorem 13.5 shows that
If
S
Consequently, shows that (4)
satisfies
a.c.c.
S = <s>^ for some (3)
implies
— > (1) :
If
a Euclidean domain.
(4)
D
S
is
D[S]
it follows is not a
is the union of an
ascending sequence of cyclic submonoids. S
S
is finitely generated since
(Theorem 7.7).
is cyclic —
implies that
isa Dedekind
But Theorem 13.7
on cyclic submonoids. s , and
S ^ Zq .
This
.
is a field,
thend [z q]
= D[X]
is
Since a quotient ring of a Euclidean do
main is again Euclidean,
it follows that
D[Z]
is also
Euclidean. Section 13 Remarks A brief account of the theory of Priifer domains is given in Section 1— 6 of [83]. Chapter IV of [51]. shows that the domain D
For a more detailed account,
see
In particular, Theorem 24.3 of [51] D
is a Priifer domain if and only if
is integrally closed and
(a,b)
2
2
2
= (a ,b )
for all
169 a,b e D ; we have used this characterization of Prufer d o mains several times in Section 13. Our restriction to integral domains in Section 13 will result in some duplication of results in Section 18, where the problems of determining conditions under which
R [S ]
a Prufer ring, a Bezout ring, etc.
On the
are considered.
is
other hand, a more complete treatment of ring— theoretic aspects of the problem are included in Section 18, for the classes of Prufer and related rings are not as familiar as their domain counterparts. Another class of domains frequently considered in con nection with the classes of Prufer domains and Dedekind d o mains is the class of almost Dedekind domains, defined as follows. main
An almost Dedekind domain is a unitary integral d o
D
such that
Dp
each proper prime ideal
is a Noetherian valuation domain for P
of
D .
Dedekind domains are al
most Dedekind and an almost Dedekind domain is a Prufer domain.
Except for the class of fields, almost Dedekind d o
mains are one— dimensional. domain
D
is Noetherian
A one— dimensional almost Dedekind
(hence a Dedekind domain) if and
only if no maximal ideal of
D
is idempotent.
Another char
acterization is that the almost Dedekind domains are the domains in which the cancellation law for ideals holds — is,
AB = AC
and
A * (0)
implies
B = C .
that
Gilmer and
Parker in [59] determine necessary and sufficient conditions for
D[S]
to be an almost Dedekind domain, but their proof
requires specialized results about such domains that are out side the scope of this monograph.
In Corollary 20.15,
however, we determine those group rings
D[G]
that are
170 almost Dedekind. D[G]
That result is strong enough to show that
need not be Noetherian if it is almost Dedekind;
example,
Q[Q]
for
is such a domain.
For a brief account of the topic of divisible abelian groups, the interested reader should consult Chapter IV of [47].
171 §14.
Monoid Domains as Factorial Domains
As in Section 13, we assume throughout this D
is a unitary integral domain and that
cancellative monoid. monoid domain
is factorial.
factorial if and only if J
is a torsion— free
We seek conditions under which the
D[S]
is satisfied in
S
section that
.
J
A unitary domain
is a GCD— domain and
J
is
a.c.c.p.
Therefore one approach to the problem
of determining conditions under which
D[S]
is factorial
would be to determine separately conditions under which is a
GCD— domain or satisfies
results together.
a.c.c.p.
This is close to
The hedging is on the
a.c.c.p.
,then mesh
D[S]
these two
the approach we
follow.
part, but the first step in
the proof is to determine necessary and sufficient conditions in order that
D[S]
condition is that operation on
should be a S
should be a
GCD— domain.One necessary GCD—monoid.
S is written as addition, a
the notation
of Section 6 is required;
valent form of the definition of a additive notation, u e s
such that
Since the translation from
for example, one equi
GCD—monoid, using
is that for all
s,t e
S , thereexists
(s + S) n (t + S)
= u+
S .
The first two
results come easily. THEOREM 14.1. GCD— domain and Proof.
S
If
D[S]
is a
Since
D
is a
is an algebraic retract of
that
D
S
take
s,t € S
such that
GCD-monoid, (Xs)
divisible by
D
is a
GCD—monoid.
Theorem 12.2 shows is a
GCD—domain, then
n (x*) = gD[S] g , part (1)
is a
D[S]
,
GCD—domain. To prove that and let
g e D[S]
. Since the monomial
of Theorem 11.1 shows that
be X s+t g
is
iii is a monomial, necessarily of the form unit of
D .
Thus
(Xs) n (X1) = (Xw ) .
v e S , is the semigroup ring (s + S)
uX
D[v + S]
, where Since
is a
(Xv ) , for
, it follows that
n (t + S) = w + S .
THEOREM 14.2.
If
torsion— free group, Proof.
Before
quotient ring
JT
D is a
then
GCD— domain and G
D[G]
of a
GCD— domain is again a
aJ n bJ = cJ
it follows that
a
giving the proof, we observe that each
aJT
and
Without loss of generality we assume that Then
is
is a GCD— domain.
To prove this, take principal ideals
J .
u
for some
a
c c J .
GCD-domain. bJT and
of b
JT .
are in
Extending to
aJT n bJT = (aJ n bJ)JT = cJT , so
JT
, is a
GCD-domain. To prove that ments
f = E^f\Xsi
D[G]
is a
GCD-domain, take nonzero
, g = Ejg^X^
of D[G]
, and let
e le
H
be
the subgroup of G generated by the set {s ^ , . . . ,s^t^, .. v Then H ^ Z" for some k , and D[H] is isomorphic to a quotient ring of the polymonial ring nomial ring over the
D[X^,...,X^]
GCD-domain is again a
result in the preceding paragraph,
GCD-domain. h e D[H]
.
Therefore
fD[H]
.
GCD-domain,
A poly and by
D [H ] is also a
n gD[H] = hD[H]
for some
It then follows from Theorems 12.1 and 12.3 that
fD[G] n gD[G] = hD[G]
.
This completes the proof of Theorem
14. 2. In Theorem 14.5, we establish the converse of Theorem 14.1.
The proof of (14.5) uses two preliminary results, plus
a couple of new concepts. D\ {0} , and if
x
If
A
is a nonempty subset of
is a nonzero element of
D , we say that
.,tm > .
173 x
is prime to
alently,
A
x
if
is prime to
THEOREM 14.3. and let N .
aD n xD
Let
A
= axD
for each a
if xd:a = xD
e A , equiv
for each
a e A .
Nbe amultiplicative system
T be the set of elements of
D
in
D
that are prime to
Assume that the following two conditions are satisfied. (1)
Each pair of elements of
multiple in (2)
D can be expressed in theform
is a GCD— domain, then
Proof.
Since the elements of
follows that aGCD—domain, b =
tDXT n D N
,c = n 2 t 2
’ w ^ ere
Moreover,
n t 2D^
T
are prime to t eT .
n i»n 2 6 N
we show that
bD
*
n cD .
If
and
t
(n^) n (s) = (n^s)
Hence x e bD
s
e
t l ,t2
and
c T . To
T .Then
, so that
Since
ru|m
D .
complete
for
n cD ,we write n^|ms
for
is In fact,
the proof, then
i = 1,2, x
Let
as
ms ,
i = 1,2,
; consequently,
and
and n|m
.
the other hand, xDXTn D = sD s b D xl n cDXT n D = tD , N N N so that
t|s
.
ntD , as asserted.
It follows that
is
.
D^ and
t e t.DXT , l N
nt e n^t^D
D
€ T *
c D N = t 2D N
a principal ideal of
ncD = ntD .
n D =
To show
is a principal ideal of
tlDN n t2DN = tDN^or some
N ,it
and write
an(*
bD^ = t^DN
therefore the extension of
m £N
is a GCD—domain.
take nonzero elements b,c e D
We ha ve
where
D
= tD for each
n = l c m { n 1 ,n2 > .
ntD c bD
nt
n e N ,t e T.
If
t e
has a least common
D . Each element of
for some
N
nt|ms = x
and
bD
n cD =
On
174 Assume that Then
S
D
is a
GCD— domain and
S
is a GCD—monoid.
induces a natural grading on the domain
homogeneous elements of nomials
D[S]
dXs . To prove that
to apply Theorem 14.3 to homogeneous elements of
under this D[S]
S-grading are the m o
is a GCD-domain, we intend
D [S ] , where D[S] .
D[S] , and the
N
is the set of nonzero
The next result paves the way
for the application of (14.3) in the setting indicated. f = Z^a^Xsi is the canonical form of a nonzero element then let
d = gcd(a^}^
up to associates
in
vertible element of
and let D
S .
and
s= gcd(s^}^ ; d
If f e D[S],
is determined
s is determined to within an in
We call
dXs
the homogeneous content
(h— content) of_ f ; it is determined up to a unit factor in D[S] .
If
f has
h— content
1 , we say that
ly primitive (h— primitive) . Clearly
f
is homogeneous -
f = dXsf* , where
f*
is
h—primitive. THEOREM 14.4. is a
GCD-monoid.
Assume that Pick
D
is a GCD-domain and that
a,b e D \{0} , s,t e S , and let
lcm{a,b} , u = lcm{s,t) . Assume that tive.
h-primi-
(a) n (b) = (c) , where parentheses denote the ideal
D[S] generated by the given element. (2)
(Xs) n (X1) = (Xu ) .
(3)
(a) n (Xs) = (aXs) .
(4)
(aXS) n (bX1) = (cXU ) .
(5)
fg
(6)
(aXs) n (£) = (aXsf) .
Proof. (2): D [(s
are
The following statements are valid. (1)
of
f,g e D[S]
c=
+ S)
is
h— primitive.
Statement (Xs ) n (Xt )
n (t + S)]
=
(1)
follows from Theorem
= D[s + S]
n D[t
+ S] =
D [u +S] = (XU ) .
12.3.
S
175 (3):
(a)
n (Xs) = aD[S]
n D [ s + I] = aD[s + I] =
a(Xs) = (aXs) . Statement
(4)
follows from
It follows from
(4)
geneous elements of multiple in
D[S]
conclude that dXw , where (3)
Applying part
d = gcd{a,b}
(4)
and
. We use these
let
< be a total order on
< s0
= 1 , then
ponent of
fg
not divisible by
gcd{w,fi>
* 1 .
Since
is h— primitive,
For
1 < i < n
r
w^ = 1 .
that
, we
D[S]
and
and
To prove that
fg °
r
Then
fails todivide some
for j
for
= 1,2,...,m .
w . , let
a previous case, and if
If
com
Thus, assume that w. = Choose
gcd{w,f^,...,f^} . i ^ 2
minimal
is a nonunit homogeneous
therefore assume thatw
gcd{w,f^} = 1 .
If
is a homogeneous
w , and it suffices to show that
minimal so that
Thus,
g , where
f^ = a^X51
w = 1 . n
divide some homogeneous component Wi-i
exist
f = Z^a^X5*
of f
g^ = b^X^j
gcdCwjf^}
divisor of
part
it suffices to show that each nonzero, no n
homogeneous component of
such that
. Thus,
andlcm{Y}
t, < t0 < . .. < t . 1 L m
w
f
and is equal to
observations in proving (5) .
unit homogeneous element
i = 1,2,...,n
D[S]
w = gcd{s,t}
g = E^bjX^ be the canonical forms i
homo
of nonzero homogeneous elements of
D[S]
s
.
of Theorem 6.1, we
exists in
of (6.1) then implies that Y
(3)
has a homogeneous least common
gcd (aXs ,bX^ }
for each finite set
and
that each pair of nonzero
D[S]
.
(1) , (2) ,
of
fails to
fg . Replacing
divides
w
f^,...,^!
by and
gcd{w,g1 > = 1 , we're finished by gcd{w,gi> * 1 , we choose
gcd{w,g^,...,gj} = 1 .
Replacing
j > 2 w
by
176 gcd{w,gj,...,g j _p w
, it suffices to consider the case where
divides each of
gcd(w,gj } = 1 .
f1 ,...,fi_1 ,
The
and
(s^ + t^)— component of
gcd(w,fi> =
fg
has the
v < i
or
form figj + (a sum Since
w
of terms fy£y > where either
divides each such
it follows that of
fg .
w
To prove
g = aXsk =
fq .
Let
h— content of k*
and
q*
q^ • q*f , where and
q*f
are
(5) .
(6) , we show that if
(aXs) n (f) ,
where
gcd{w,f^gj}= 1 ,
(s^ + t^)— component
the proof of
element of
be the
anc* since
fails to divide the
This completes
(6):
fy gy
then
and
q^
h— primitive.
differ by a unit factor in
write k
k = k^k*
h— primitive.
aXsk^
is a nonzero
h— content of
q , and write are
g
g e (aXsf) .Thus,
k^ be the
y < j).
,let
and
q = q^q*
Then aXsk^
It follows that , so
f|k*
k* .
,
• k* =
are homogeneous and
D[S]
q^
k*
and
Hence
q*f g =
aXsk xk* e (aXsf) . THEOREM 14.5. GCD-monoid, Proof.
then
If
D
D[S]
is a
GCD-domain
and
S— grading on
be the set of nonzero homogeneous elements of be the set of elements of
D[S]
h— primitive elements of
f = ‘aXs f*
D[S]
T
(f):aXs = (f*) * (f)
a
so that
D[S] .
.Part
(6)
h— content f
, let Let
N .
We
are precisely the
h— primitive element of has nonunit
D[S]
that are prime to
show that the nonzero elements of
And if
is
is a GCD-domain.
Under the canonical
shows that each
S
of Theorem 14.4 D[S]
is in
T
.
aXs , then
is not prime to
N .
We
N T
177 note that conditions isfied for part
(4)
Moreover D
and
that
N
and
(1)
and
T ; that
(2) (2)
of Theorem 14.3 are sat is satisfied is clear, and
of Theorem 14.4 shows that D [S }^ ~ K[G]
G
, where
K
is a
is satisfied.
is the quotient field of
is the quotient group of
D[S] n
(1)
S .
GCD— domain, and hence
Theorem 14.2 shows D[S]
is a
GCD— d o
main by Theorem 14.3. We turn to the problem of determining conditions under which
D[S]
is factorial.
necessary conditions on lish.
As with the
D
and
S
concerning
(1)
If
satisfies ticular,
Let
a.c.c.p.
D
K
be the quotient field of D .
is satisfied in
for each subgroup
satisfies
If
satisfies
14.6.
a.c.c.p. D
(2)
and
H
K [S ]
satisfy
S .
a.c.c.p.
Statement
(1)
In pa r
, then D[S]
follows from Theorem 12.2.
(2) , take an ascending sequence
i .
of
a.c.c.p.
of nonzero principal ideals of Since
K[S]
g^ is a unit of
trivial (Theorem 11.1), d^Xsi , where
D [S ] —
satisfies
out lossof generality that
form
D [S ] , then D[H]
a.c.c.p.
Proof.
each
We first prove a result
a.c.c.p.
THEOREM
each
are fairly easy to estab
Theorem 14.7 represents an expanded analogue, for
factorial domains, of Theorem 14.1.
prove
GCD—property, some
Cf^) say
a.c.c.p.
£ (f2^ £ • • • f^ = g^f^+i
.
Since units of
it follows that each
s^ is invertible in
S
for
, we assumewith
f^K[S] = f2K[S] = ...
K[S]
To
g^ .
.
Thus,
K [S ] are is of the Let
a^
be a
178
nonzero coefficient of some coefficient
a2 ^ 0T
f^ = d^X5! ^ of
f2 •
a2 =
have
a^D c a 2D £ ... , and since
some coefficient
D , it follows that
d^Xsi
is a unit of
...
This completes the proof of
.
THEOREM 14.7. let
G
then
D[S]
Let
for
K
and
S
for some
D .
k .
Conse
Therefore (f^)
=
=
be the quotient field of
Moreover,
We
=
(2).
S .
are factorial and
cyclic submonoids.
f^ , etc.
is satisfied in
i > k ,and
be the quotient group of D
of
a.c.c.p.
are units of
^ 0T
f2 = d 2X S2fj ,
a^
a^D = a^+1D = ...
d^,d^+ i ,...
= ^ia2
Then
And since
then
quently,
.
If
S
D [S ]
and
is factorial,
satisfies
D[G]
D
and K[S]
a.c.c.
on
are factorial
domains. Proof. Theorem 12.2.
Since
D[S]
is factorial,
To prove that
invertible element
S
s e S .
Each
JL
where Then of
u. l
D
and p. *i
X s as a , say
take a non-
finite product Xs
= - --IIn .
of the form
u^X^ 1 ,
is not invertible in
S
.
s = p^+ -«‘+Pn > and to complete the proof of factoriality S , we show that each
a,b e S
are such that
which implies either and
D[S]
is a monomial, necessarily is a unit of
is factorial by
is factorial,
Write
of nonunit prime elements of
D
p^
a + b e p^ +
X*
b e p i + S . Therefore
pi
S .
is prime and
Moreover,
S K[S]
because they are quotient rings of COROLLARY 14.8.
S .
e (XPi ) and aepi + S
Theorem 13.7 implies that cyclic submonoids.
is prime in
Thus, assume
Then or
X aXb
X** e (XPi )
S is factorial.
satisfies and D [G ]
a.c.c.
on
are factorial
D[S] .
Under the notation
e (X^i),
of Theorem 14.7,
179 D[S]
is factorial if and only if Proof.
D and
K[S]
are factorial.
In view of L14.7), we need only show that
is factorial if
D
and
K[S]
are factorial.
Thus,
D[S] S
is a
GCD— monoid by Theorem 14.5, and the same result shows that D[S]
is a
part
(2)
GCD—domain.
Since
of Theorem 14.6,
a.c.c.p.
holds in
it follows that
D [S ]
D[S]
by
is fac
torial. It turns out that the three necessary conditions and
on
S given in Theorem 14.7 are also sufficient for
toriality of proof.
D[S]
D
and
S
are factorial and let
the group of invertible elements of S
fac-
, but additional work is required for a
Assume that
know that
D
is isomorphic to
ternal weak direct sum of
a
S .
H
be
From Section 6 we
H © M , where copies of
M
is the ex
, with
a
the
cardinality of a complete set of nonassociate prime elements of
S .
Hence
D[S]
ring over
D
Therefore
D[M]
- (D[M])[H]
is factorial,
should be factorial.
monoids.
D[M]
is a polynomial
in a set of indeterminates of cardinality
H
a .
and we seek conditions on
in order that the group ring of
condition on
, and
H
H
over a factorial domain
According to Theorem 14.7, a necessary
is that it satisfies
a.c.c.
on cyclic sub
The next two results are concerned with this
condition in an abelian group. THEOREM 14.9. Let
Assume that
G
is a group and
M be the set of positive integers
is solvable in (1)
G
(that is,
The set
M
g
n
g e G .
such that
is divisible by
n
nx = g in
G)
.
is closed under taking positive divisors
and under taking least common multiples.
180 (2)
The following conditions are equivalent. (i)
M
is finite.
(ii)
M
contains a largest element
(which is
is necessarily the least common multiple of the elements of
M) (iii)
M contains only
and for
each
p e M , there is a largest power
p
is in M
that
(3)
If
M
finitely many primes
p
p
of
.
is finite, then
the element
g
has
infinite
order. Proof.
(1):
If
each positive divisor
n e M , it is clear that d
of
n .
each prime-power divisor of show that
lcm(n,m}
case where
n
nx = my = g
and
is in m
and that
and
1 = na + mb .
n,m € M , then
is in
M .
Hence, to
Then
Assume that g = nag + mbg =
.
The implications
(i) ==> (ii) (iii) ~ >
and
(ii) = >
(iii)
(i)
follows from
• To prove
then
for
M , it suffices to consider the
are patent, and the implication
(1 )
m
if
are relatively prime.
namy + mbnx = mn(ay + bx) (2):
n
Thus,
d e M
(3) , we note that if
(g) = (rg)
prime to
k
g
has finite order
for each positive integer
so that each such
r
is in
M
r and
relatively M
is not
finite. If
G
and
g
are as in the statement of (14.9), then
the terminology of abelian group theory is that type
(0,0,...
)
if the set
M
g
is of
of Theorem 14.9 is finite.
To a reader seeing this concept for the first time, the
k ,
181 choice of terminology must seem abstruse, but in fact, the type of an arbitrary element
h e G
can be defined (see
[48, Section 85]); we have no occasion to use the general concept of type. between type
The next result indicates the connection
(0,0,... )
THEOREM 14.10.
and factoriality of
Assume that
G
D[G]
.
is a torsion— free group.
The following conditions are equivalent. (1)
G
satisfies
a.c.c.
on cyclic submonoids.
(2)
G
satisfies
a.c.c.
on cyclic subgroups.
(3)
Each rank— one subgroup of
(4)
Each nonzero element of
Proof.(1) < = = > (2) : follows since tains
G
G G
properly contains
t = ns (t)
is satisfied and let
is of type
The implication
is torsion— free:
^ , then
is cyclic.
.
if
for some
(2) = = > ( 1 )
<s>^
properly con
n > 1 , and hence
(s)
For the converse, assume that
(1)
(g^) £
- **'
quence of nonzero cyclic subgroups of g l = n lg 2 =
(0,0,... ) .
n i)(~ 8 2) •
be an ascending se G .
We have
Hence by replacing
if necessary, we may assume that
n^
g2
is positive.
- g2
Similarly,
after a possible replacement of
g^
that
Without loss of generality,
g 2 = n 2S 3
assume that
(gk+i ) =
n2 > 0 .
g. = n.g. , 6i is i+l
^ 9 we have
D[G]
n W* =
Therefore D[G]
f ,
Each nonzero element
X af*/X^g*
are not in
It follows that
X gh
.
, it suffices to show that
is expressible in the form
f* € D[W]
ment of
D[S]
For
with respect to
U = {X1* | t € T}
is essential for
.
g = (g + t) —
W c y , and the reverse inclusion is clear.
converse, we note that
is
f e D[S] , h $ D[S\T]
n T . From the equality
longs to the quotient monoid
w*
w* .
for some
t e Supp(h)
.
with value group
S , and the assertion concerning the center of
(2) :
on
L
holds since
immediate from the definition of
of
k
n (naW*) = naD[Wa] = D[nWa] = D[S]
.
,
201
Assume that Z?aiXSl
{w a )
has finite character and let
be the canonical form of a nonzero element
f e
D[S]
the
set B= (a e A |wa (si)
n)
is finite, and
{w*} a
f =
.Asin the proof of part * 0
w*(f) = 0
(3)
of
forsome
i
between
a
for each
Theorem 15.3,
e
1
and
A\B .
Thus
if
is a
is also of finite character, If G
is the quotient group of
valuation on then we say
G
respect to
and
whose valuation monoid
that
essential for
S
W
w is essential for
S)
if W
is the
S
w
contains
S ,
(and that
W
is
S
with
quotient monoid of
T = {s e S | w(s) = 0} .
The next result
follows
from Theorem 15.7. THEOREM 15.8. quotient group of type
G
Assume that
S
is a Krull monoid with
and that each nonzero element of
(0,0,... ) .
^ct^aeA
Let
S
character.
.
Then S = ^ a e p ^ a > and
If
^U3 ^ 3 €B
valuation monoids on n$eBus * then
< V
£
G
{V
that are essen {Wa >
*s any fami!y
G
is
°^
rank-one discrete valuation monoids on tial for
G
has finite
°f rank-one discrete
of finite character such that
S =
•
Proof. Let w be a valuation associated with W , u ----a a 3 a valuation associated with ,and let K be a field. If F
is the defining family forK[G]
Theorem 15.7 shows that for the Krull
domain
is
r
F u {U*}
S) .
i
Y
We have
, then
F u K[S] . The
, where
part
of
is a defining family defining family for
C = {3 e B | U*
p
is essential
K[S] = K[G] n (ny£CU*) = K[ny UY ]
proof of Theorem 15.7), so that
(3)
S =
•
K[S] for
(see the
202
)E
*s clea r*
On the other hand, each
W*
essential rank-one discrete valuation overring of hence belongs to the defining family of
W*
on
D [S]
is
D[S\Ta] , where
This ideal extends to K[G]
K[G]
for each
s e S \Ta
W* = U*
for some
hence
k[ G] n U* = K[U ] , so L
y
F u {U*} .
.
a
, for
Xs
It follows that
W*
= U
Thus
.
D[S]
, and
The center
T a = {s € S |w a(s) * 0} .
K[G]
e C .
Y
W
Y
in
is an
K[G]
is a unit of { F , and W* = K[Wa] =
n
We conclude that
{U } = Y
y
(W^} , and this completes the proof. COROLLARY 15.9.
Assume that
The defining family for
{ K rG V
)
D[S]
D[S]
is
}i £I U {D[S]P [ S ] }a eA u {W6 }6£B > a
where the notation is as follows: of
D ,
G
is a Krull domain.
K
is the quotient field
is the quotient group of
S ,
is a com
plete set of nonassociate prime elements of
K[G] , and
{Pa )aeA
is the set of minimal primes of
{Wp}peB
is the family of rank-one discrete valuation monoids
on
G
that are essential for
D ; finally,
S .
Several remarks on the statement of Theorem 15.8 and Corollary 15.9 seem appropriate. valuation monoids on following reasons. on
G
Note that ments of
W
of
ker(w) W .
Thus
(15.8) is stated for
G , rather than valuations, If
w
with value group
tion monoid
First,
w
is a rank-one discrete valuation dZ , where
is
for the
d > 0 , then the valua
ker(w) © ^ » where
w(g) = d .
is precisely the group of invertible ele W
does not determine
rather up to the value of
w
on
g .
w
uniquely, but
This wee difficulty
203 can be avoided by assuming that all rank-one discrete valua tions dealt with have value group said to be normed) .
Z
(such a valuation is
Working with normed valuations, Theorem
15.8 can be rephrased by replacing and
(up)
, respectively.
neither of the families
D [S ] p j-gj
(K[G] ^
^ ^ ei
’ a description in terms of
tained:
K[ G] (q.} = D[S ](q.)nD[s]
(S\Tg}geB If of
S
{wa )
.
is described
D[G]p
.
With
D[S]is easily Similarly,
ob-
W* =
Theorem 16.10 of Section 16 we show that
is the set of minimal prime ideals of S
by
; in the other d i
could be written as
] > aRd
(Ug)
or
D[S]
K[G](q.)
D[S] d [s \t
and
In the statement of Corollary 15.9 ,
implicitly in terms of the domain rection,
{Wa )
is finitely generated,
is also finitely generated.
S .
then the quotient group
G
The converse need not
hold in general, but we show in Theorem 15.11 that the con verse holds if
S
is a Krull monoid.
An ancillary result
is used in the proof of (15.11).
THEOREM 15.10. Let A be an infinite subset of v where F = Zl^Ze^ is a free group of finite rank k . by
1 , let M =
n^(A)
be the
set of first coordinates of elements of infinite for some
F+ ,
A .
If
Il^fm)
is
m e M , then the induction hypothesis implies
JI^1 (m) , and hence
* increasing sequence.
A , contains an infinite strictly
On the other hand,
if
-1
(m)
is finite
204
for each A
m € M , then there exists an infinite subset
such that distinct elements of
n i ^ 2^ < ' ’ ’ *
If
Let
B
ci = ^i ~~
f°r each
is a finite set, then there exists
such that the set {bi>iei
of
have distinct first oo Thus, we may assume that B = { b ^ ^ , where
coordinates. ni^l^
B
I = {i e Z+ | c. = c}
i •
c e {c^}”
is infinite, and
is aninfinite strictly ascending sequence in
In the contrary case, where
A .
is infinite, the induction
hypothesis yields an infinite strictly ascending sequence c^
< c^
< ...
of the set
.
Let
{II^ (b
^
the smallest integer Then
= i^ .
b. < b. J1 J2
Only finitely many elements
are less than i^
) .
such that
and we can
ni ^ i t ^
We let
jT j
II-. (b. ) > n, (b. ) . 1 ^-u 1 J2
{
be the smallest
Then
j2
OO Cb ^ a r e i u
be
(b . ) .
>
repeat the argument on
wit, only finitely many elements of II^ Cb. ) . 1 j2
Let
b. .To J2 less than
such that
b. < b. < b. , and a continuaJ1 ^2 J3
tion of the process yields an infinite strictly ascending sequence
(b. } Jr
in
A .
The result follows by the principle
of mathematical induction. THEOREM 15.11.Assume quotient group then
S
S
is a Krull monoid
with
G is finitely generated as a group,
is finitely generated as a monoid.
Proof. family
G .If
that
Assume that
G
is generated by
{wa^a€A °f nontrivial valuations on
character is necessarily finite since finite for each
i
defining family for
between S
1
and
is finite.
{g^ G
.
A
of finite
{a £ Alw^fg^) * 0 } n .
is
In particular, a
The proof of Theorem 15.2
205 shows that
S
is of the form
of invertible elements of where
F
H © T , where
S
and T
F . Since
H
s G, H
as a group and as a monoid.
is of the
shows that the set Let
U
B
be the submonoid
less than
t , so
t e U .
Consider
If
b^ . If
other case, T .
F
of
T c U.
Thus, take
t * b^
, then
t > t -b^
t - b^
t > t — b^
and
. To
a
non
F+
M n F+ = such that
= b2 , w e ’re finished, and
in the
> t - b ^ — b 2 is a decreasing
sequence
Again because only finitely many elements of
Therefore
T = U , and this completes the proof. T
are
t = b^ ,
t
b2 e B
is
B
for some element b^cB . I f
t , it follows that
free on
T
Only finitely many elements of
t > b^
T
Theorem 15.10
T generated by
less than
If
is a sub
as a partial
< .
Repeating the process, there exists
b2 < t -
in
t e T .
M
of nonzero minimal elements of
complete the proof, we show that zero element
M n F+ ,
Hence, it suffices to show that
ly ordered group under the cardinal order
T .
form
group
is finitely generated both
is finitely generated as a monoid.
then
is the
is a finitely generated free group and
group of
finite.
H
is of the form
M
t = b-^ +
...+b^ € u
n F+ , where
{ea ^aeA » then the monoid
domain
garded as a subring of the polynomial ring
F+ are forsome
F = ^ 0LepiZea D[T]
is
can be
D[{X }
*]
Ot Ot € A
k.
re over
D .
Moreover, D[T] is generated as a ring over D by "pure 0 6 e monomials” X *X 2 ...X n , with e^ £ 0 for each i . Conal a2 an versely, each ring
, where each
monomial in the indeterminates
Xa , is of the form
where
U
is a submonoid of
m^
is a pure
F + . Rings of the form
t>[U] , D[{m^}]
arise as objects of natural interest in various parts of commutative algebra.
Thus,
it seems worthwhile to state the
206 following consequence of Theorem 15.11 as COROLLARY 15.12.
Assume
that
D is
grally closed domain and that indeterminates over
D .
Let
{ma ) » and let
separate result.
a Noetherian inte a
{m a }aeA
monomials in the indeterminates generated by
a
finite set of 3 S6t
X. , let l *
T
J = D[{ma >] .
Pure
be the monoid The following
conditions are equivalent. (1)
T
is finitely generated and integrally closed.
(2)
J
is Noetherian and integrally closed.
(3)
J
is a Krull domain.
We remark that under the notation of Corollary 15.12, integral closure of
T
is finitely generated. D +XD[X,Y] each
does not, in general, For example,
imply that
D[{XY1}~= q ] =
is a non-Noetherian integrally closed
integrally closed domain
T
domain for
D .
Section 15 Remarks Three general references on Krull domains are [20, Ch. VJI], [44], and [51].
A number of other classes of domains that
are related to Krull domains have been considered in the literature.
Among classes whose definitions involve certain
conditions on the family of valuation overrings of the given domain, there are the classes of domains of finite character, domains of finite real character, domains of finite rational character, domains of Krull type, and generalized Krull do mains
(for the definitions, see Section 43 of [51]).
Each of
these concepts has a natural extension to monoids, so that one could consider the classes of (always torsion— free and
207
cancellative) monoids of finite character, etc.
Using
Theorems 15.3 and 15.7 and other techniques from this section, it can be shown that the monoid domain named classes if and only if S
D[G]
D[S]
is in one of the
belongs to the class, and
belongs to the monoid class of the same name; here
notes the quotient group of
S .
0 , Matsuda in [97] has shown that if and only if of type
D
(0,0,...
For
D
D[G]
G
of characteristic is in a given class
is in the class and each element of ) .
For
D
de
of characteristic
G
is
p > 0 , the
conditions are almost the same; the difference in this case is that divisibility of nonzero elements of powers of
p
G
by arbitrary
does not affect whether or not
D[G]
belongs to
the class (see [100] for details). Another class of domains related to Krull domains are the
n—domains, where
D
each principal ideal of ideals of
D .
II— domain iff invertible iff D
D
is defined to be a D
is a finite product of prime
According to Theorem 46.7 of [51],
is a
is factorial for each maximal ideal
Anderson and Anderson
D
D
M
of
are finitely generated.
[5] have shown that
n— domain if and only if
is a
D[S]
II—domain,
and each element of the quotient group of ) .
D
is a Krull domain whose minimal primes are
and minimal primes of
(0,0,...
n— domain if
S S
is a is factorial,
is of type
208 §16.
The Divisor Class Group of a Krull Monoid Domain
Theorem 15.6 gives necessary and sufficient conditions for a monoid domain
D[S]
to be a Krull domain, and subse
quent results in Section 15, such as Corollary 15.9, determine the defining family for primes of
D[S]
.
D[S]
and the family of minimal
Using heavily these results from Section
15, we determine in Corollaries 16.7 and 16.9 the divisor class group of
D[S]
.
To begin, we review some of the basic
terminology, results, and notation concerning the
v-operation
on an integral domain, and in particular on a Krull domain. Later in the section we replicate much of this development for cancellative monoids, assuming no familiarity on the part of the reader with the results in the context of monoids. Let K of
D
be a unitary integral domain with quotient field
and denote by D .
For
F(D)
the set of nonzero fractional ideals
F e FCD)
,F~*
is defined to be
{x e K | xF c D} , and the fractional ideal noted by
Fv .
Equivalently,
Fy
F -- ► Fv
is called the
and a fractional ideal v— ideal)
if
A v = Bv .
div(A)
noted by
that contain
v— operation
of
on
v—operation on on
F(D)
V (D)
, and
.
D ,
(or a
defined by setting A ~
D , the class of
Under the operation
~
are called
A e F(D)
is denoted D
div(A) + div(B) =
is a group if and only if
D
is
B
is de
is a commutative monoid with zero element P(D)
F .
Dinduces an
, and the set of all divisor classes of
div(AB) , V (D) div(D)
D
The equivalence classes under
divisor classes by
~
is de
is said to be divisorial
F * Fy . The
equivalence relation if
F
(F *) *
is the intersection of the
family of principal fractional ideals of The mapping
D:F =
209 completely integrally closed. {div(xD) | x e K,x * 0} ible elements of
P(D)
F e F(D) of
C(D)
, and C(D) = D ; if
D
P(D)/P(D)
[div(F)]
determined by
div(F)
of invert
is
called the
is completely integrally
is the divisor class group of
, we denote by
C(D)
P(D) =
is a subgroup of the group
divisor class monoid of closed, then
The set
the element
D .
For
div(F) + P(D)
.
We now specialize the discussion of the preceding para graph to the case of a
Krull domain.
a Krull domain, that ideals of
Thus, assume that
D
is
the family of minimal prime
D , and that
v^
is a normed valuation on
K
associated with the valuation domain i e I .
V. = D_ for each 1 pi In this case it is known that fciivCP^D JicI
free basis for the group Fy =
> and
scripts
i .
FV^ =
Moreover,
%
P(D) .
In fact, if
a
F e F(D)
, then
forall but a finite set of
for an integral ideal
A
sub
we have
■
is a finite intersection of symbolic powers of the minimal primes
P^
of
D
that
contain A .
The first
section relates the divisor class group of quotient overring THEOREM 16.1. Krull domain F(D^)
D .
induces
The kernel of prime of
D
Proof.
of Let
N
be a multiplicative
The mapping
F -- ► FDN
is generated by
that meets
to that
of
a
D .
a homomorphism <J>
D
result of the
of
C(D)
of
system in the F(D)
onto
into
C(Djyj) •
{[div(P)] | P is a minimal
N} .
Consider the mapping
a:P(D) -- »■ P(DN )
defined
210
by
a(div(F)) = div(FDN )
for
F e F(D)
is well-defined, we first show that exists an integral ideal of
D
such that
d 1 (ADn )v . Ay D^ .
A
of
F = d 1A .
Hence
meets
\
=
j .
contain
v
= d *A
v
containing
A
meets
of minimal primes of
nT=1pjn P
D
containing
{pjDN^i
, where
ADp
= PjjDp
=
= nJ c P j V (nj) •
the set of minimal primes of
P;iDp.= CPjDp,)n j CDN )p.n
j
div(F) = div(G)
J
j
1
, then
j
j N
= AyDN .
Fy = Gy ,and hence
is well-defined.
It is clear that
Thus,
if
div(FD^) =
a
This shows
preserves
is surjective since each integral ideal of D .
Since
PCD)
free on (divCPD^)
follows that o
g
is extended from
and
that
for each j .
div((FDN )v ) = divCFyDN ) = div(GvDN ) = divCGDN ) .
Dn
DN
and
Therefore (ADN )y = nj=1 CPj D„) (n 35
addition, and
A
We have
j N
a
(divCP)| P
meets
|P N}
is free on does not meet
(div(P)} N} , it
generates the kernel of
. Since
,
P. does not meet N for 1 < j £ k , J J N for k + 1 < j < r . Then
(ADN )CDN)p.D = ADP.=
that
N
(ADXT) = DXI = A DXI . Thus, assume v N'v N v N *
nI (pjnj)V
ADXT N
D
F
d
(ADN )v =
is labeled so that
Moreover,
and a nonzero element
Thus, it suffices to prove the equality
{P^
a v dn
There
(FDK1) = N v
that the set
for each
*
a
and
then it is clear that
Pj
To show that
(FD^)v =
D
If each minimal prime of
while
.
a(PCD)) c PCDn ) , then
a
induces a surjective
211
homomorphism
<J>: C(D) -- »- C(D^)
[divfFD^)]
.
Suppose
Then
= yD^
hence
[div(F)]
so
divfy *F)
divfy ^F) = d i v ( P ^ .. .pjjm )
{ki>i .
D
meeting
It follows that
that meets
N}
e ker a
for some set
N
<j> .
, and
{P^
of
and some set of integers
[div(F)] = [div(P^1 ...P^m ) ] =
Z^ki[div(Pi)] , and hence D
, as asserted.
There are strong similarities between the proofs of Theorem 16.1
and the next result, which relates the class
group of a Krull
domain
THEOREM 16.2.
D[S]
to that of
D .
Assume that the monoid domain
D[S]
is a
Krull domain. (1)
If
A is a nonzero integral ideal of D ,
(A[S])y = A y [S] (2)
. F -- ► FD[S]
The mapping
of
induces an injective homomorphism C(D[S])
If
S is a group, then
C(D[S]) = C(D) Proof. D
of
into
C (D)
F(D[S])
into
(1):
and
Letbe the set of minimal primes
that contain
A
and for
1 £ i £ k , let
K , the quotient field of
with the valuation ring k (Pits]}!
is surjective
.
normed valuation on
that
<J>
F (D)
.
(3)
of
then
V- = D p
.
v?
associated
D[S]
contains
A[S]
D[S]p [sj
, defined as in the statement of Theorem 15.3
have
Let
D,
be a
Corollary 15.9 shows
1 is the set of minimal primes of .
v^
that
be the valuation associated with We
212
A v = n£_1
^
, where
(A[S])y = n^(Pi [S])
,
nu = inf(v^fa) | a e A) , and where
n i = inf{v?(b)| b
It is clear from the definition each
i .
of
v*
that
e A[S]}
n^ = nu
for
Since
A yD[S] = (n^pfHi^DfS] to complete the proof of
(1)
= n^CP^n i ) [S])
(P^[S])^n i^
from part
(2) of Corollary 8.7:
and cancellative, the contraction to
for each
P?n i^[S] D[S]
,
, it suffices to show that
pf-n i^ [S] =
is
i .
This follows, however,
since
S
is torsion— free
P^[S]—primary , and hence is
of
P-n i ) [S] . D[ S]p.[s]= (Pj[S]D[S]p [s])n i This completes the proof of (2): [div(F
The mapping
• D[S])]
.
(1)
<J> is
Using
(1)
•
•
defined by
<J>([div(F) ])
, the proof that
(j)
=
is
well-
defined is the same
as the proof of the same assertion
a
Clearly
in Theorem 16.1.
[div(F)] F
e ker <J> .
Since
fD[S]
monomial, and since invertible in D
is a homomorphism.
for
Suppose
Without loss of generality we assume that
is an integral ideal of
principal.
to
.
D
FvD[S] = fD[S]
contains monomials,
Fv c aXsD[S]
S , and hence
then yields
. Then
Fy = aD
is
f = aXs
, it follows that
FvD[S] =aD[S] , so that
.
s
is
a
is
Contraction
[div(F)]
=
is injective, as asserted. (3):
Suppose
S
is agroup.
each nonzero element of if
N = D \ {0} , then
Theorem
S is oftype
(D[3])N
= K[S]
(0,0,...
15.6 shows that )
is factorial.
.
Hence, Thus
0and<J>
213 K[S]
has trivial class group, and Theorem 16.1 shows that
C(D[S])
is generated by
D [S]
that meets
that
D[S]
D} .
is a minimal prime of
Applying Corollary 15.9, it follows
is generated by
prime of of
N} .
{[div(Q)] | Q
{[divP^JS]] | P
Since each
<J) , we conclude that
[divPa [S]] <J>
is a minimal
belongs to the range
is an isomorphism of
C(D)
onto
C(D[S ]) . Suppose
D[S]
16.2 determines S .
is a Krull domain.
C(D[S])
in terms of
THEOREM 16.3.
S
and
S
of Theorem for a group
need not be a group.
Assume that
be the quotient field of
group of of
D
(3)
Using this result, Theorem 16.3 advances such a determi
nation in the case where
K
Part
S , and let
H
D[S]
is a Krull domain.
D , let
G
Let
be the quotient
be the group of invertible elements
S . (1)
C(D [S] ) ~ C(D) © C(K [S]) .
(2)
As
in the statement of Theorem 15.2,express
H © T , where C(K[S])
T = G n F+ , with
~ C(L[T])
Proof,
, where
(1):
Let
L
D[S]
: C(D) -* C(D[S])
a homomorphism.
that the
defined by
Moreover, part
map y: CCD) -- ► C(D[G])
[divFD[G]]
is an isomorphism.
identity map
on
summand of
Since
as
Then K[H] .
be defined as in
D[G]
is a quotient
, Theorem 16.1 shows that the mapping
y C(D[G])
oi:C(D[S])
a free group.
is the quotient field of
the statement of Theorem 16.2. ring of
F
S
C(D)
C(D[S])
—
a([divF]) = [divFD[G]]
(3)
of Theorem 16.2 shows
defined by Since
y([divF]) =
y *a<J) is
, it follows that <J>(C(G)) say
is
C(D[S]) = <J>(C(D)) © M
the is a direct .
If
214 N = D \ {0} , then as in the proof of Theorem 16.2, there exists a surjective homomorphism with kernel generated by of
from
t
C(D[S])
([divP^fS]] | Pa
onto
C(K[S])
is a minimal prime
D) ; since this latter subgroup is precisely
it follows that
<J>(C(D)) ,
C(K[S]) ~ C (D [S ])/(C(D)) ~ M .
Therefore
C CD [S]) = then
I:J^ 2. I:J2 •
Hence
CJi)y £ (J2^v * (4)
Iv is
the intersection of the family of all prin
cipal fractional ideals of
S
that contain
I .
216 Proof.
C D : Take
L n S * <J> and that
(L
n S)+ S £
is clear.
For the first, take
where
e S .Then
u,v
u e L n S . and so
To show that such that
u + a e
I:J .
so
+I E S .
b +
omitted. I v
That S + (I:J)
- x e S :I
+S 2
and
I
and take
b e S
b + v + xeb + Is
I £
I
such
s,
x +
S
principal fractional ideals
for some
s e S .
for some
but
y - x ^
S , so
follows
if and only if
x
(1)
.
implies
is contained in the interv
^ x + S
(5)
9
(3)are routine, and hence are
y
Statement
n S
, and this completes the proof of
(2)
- s + S
hence
is clear from the
I ; this family is nonempty since
I £
is such that
x
nS
Ifx e I :J , then
section of the family of
implies
and
as u — v ,
u + a + J cu +S ci
c I:J
v e J
= x + S . so *
that contain
and write it
is nonempty, choose u e I
To prove (4) , note that
£ (x + S) v
. Wefirst observe that
The second assertion
a + J eS . Then
v + (I :J) £ S
The proofs of
L.
y e L
I :J
definitions. Finally, take b
F(S)
u = v + y e S + L £ L ,
a e s
that
L e
of S
s +I
£ S
Conversely, if
x + S
containing
y
e G
I, then
y { S:(S:I) = Iy .
from
(4)
and
from the fact that
+ S 2 Iy . To
prove
(6) , we
note that Iv = n{a +
S | I £ a + S} , and hence
x +
+ n{a + S} = n{x + a + S | I £ a + S } =
Iv = x
n(x + a + S | x + I c x
+ a+S}
n{b + S | x + I c b + S } = ( x + (7)The inclusion (I
+
J)y£
(Iy
I + J £ I+ + Jy )v .Moreover,
= 1)^ implies that since
Iy + Jy£(I
+J)y ,
217 then
(Iy + Jy )v c ((I + J)v )y = (I + J)y , and hence the
equality If
S
then the F(S)
(I + J)^ = (1^ + Jy )y
is a cancellative monoid with quotient group
v—operation induces an equivalence relation
defined by
div(I) P(S)
holds, as asserted.
I ~ J
if
I
= Jy .
For
denotes the equivalence class of
under
denotes the set of all divisor classes of
of Theorem 16.4 shows that the operation fined by Under
divfl) + div(J) = divfl + J)
+ , the set
zero element
P(S)
div(S)
.
~
I € F(S)
I
+
,
on
,
~ , and
S .
on
G
Part
P(S)
(7)
de
is well-defined.
forms a commutative monoid with Moreover, the set
P(S) =
{div(x + S) | x e G}
is a subgroup of the group of invertible
elements of
The factor monoid
PCS)
.
called the divisor class monoid of integral domains, of
C(S)
.
[div(I)]
C(S) = P(S)/P(S)
S .
As in the case of
denotes the element
Under what conditions is
PCS)
div(I) + P(S)
a group?
Based
on the domain
case,the answer should be "if and only if
is completely
integrally closed".
S
some fixed element THEOREM 16.5. quotient group
G .
Recall from
is completely integrally closed if
contains each element
x
of
s e S Let
G
S
Theorem 16.5 shows that,
indeed, this is the appropriate condition. Section 12 that
is
such that
s + nx e S
S for
and for each positive integer S
n
.
be a cancellative monoid with
Then
P(S)
is a group if and only if
S
is completely integrally closed. Proof. To show that
Assume that P(S)
an inverse in P(S)
S
is completely integrally closed.
is a group, we for each
I e
show that F(S)
.
[div(I)]
has
Without loss of
218 generality we assume that in
S .
I
is a divisorial ideal contained
It suffices to show that
that is, that
(I +
(S:I))v = S .
i t ’s enough to show
that
fractional ideal
+ S
x
S of
Therefore
I e - x +I e I
over
S , so
- x
almost integral
n e z+ , where U n = 0 (ny
+
in
P(S)
so
y e S
S)
.
s e S . of
S
Since and
Suppose
over
K
•
S
- x
Then
1 +
because
field of
is a field and C(K[S])
K[I]
1
=
Let
y e G
for each
1
’
so
d
i
v
*
it follows that
K[S]
divfS)
I £ S
and
is a Krull domain.
C(S)
In
are isomorphic, one
The statement of Theorem 16.5 for a fractional ideal
for the family of elements
I need not itself be a semigroup;
I
f e K [G ]
denotes the set of elements K[S]
i t ’s easy to show,
K [I ] is a fractional ideal of K [S]
such that
THEOREM 16.6.
yK[J]
Assume that
K [I ] :K [J ] = K [I :J ] for
y
c K[I]
S
. As usual,
of the quotient
.
is a torsion— free can
cellative monoid with quotient group (1)
is almost integral
Supp(f)c I . This is a variance from usual notation
however, that K[I]:K[J]
for
be the fractional ideal
contains an aberration in notation:
such that
. Thus
S is completely integrally closed.
preliminary result is needed.
S , we write
- nx + I £ I
s + ny e S
say I
I + (S:I)
is a group.
is divisorial,
order to prove that
of
S ,
+ S , as asserted.
P(S)
S — Let
+ (S:I) e
I c S :(S:I) = Iy = I .
, and
and S E x
Conversely, assume that be
-x +
n . Hence
e S
I
S containing
impliesthat
each positive integer
Since
is contained in each principal
x + S ^ I + (S:I) -2x +
div(I + (S:I)) = div(S) —
G .
Let
I ,J £ F (S) .
K
be a field.
219 (2)
(K [I ])v = K [Iv ]
(3)
The mapping
y: P(S) --- P(K[S])
y(div(I)) = div(K[I]) .Proof. mediate. s e
J
(1):
.
is
The inclusion
yXs £ K[I]
y e K [G ] , however, yX S e K [I ]
K [I :J ] sK[I]:K[J]
implies
take
y
and each
s e J
iff
s € J
y £ X'SK[I] £ K[G]
t + s e I
iff
is im
e K [I]:K [J ] .
it is clear thaty e K [I ] :K [J ]
for each
t e Supp(y)
by
an injective homomorphism.
For the reverse inclusion,
, then
defined
If
. For
iff
for each
Supp(y) E I :J
iff
y € K [I :J ] . (2)
It follows from
(1)
that
(K[I])y =
K [ S ]:(K[S]:K [I]) = K[S:(S:I)] = K[Iy ] . (3): Iy = J y
For iff
I ,J e F(S)
, we have
K[Iv ] = K[Jy ]
div(K[I]) = div(K[J]) and injective.
.
iff
(K[I])y = (K[J])y
Therefore
Moreover,
div(I) = div(J)
y
iff iff
is both well-defined
y(div(I) + div(J)) = y(div(I+J)) =
div(K[I + J]) = div(K[I] • K[J]) = div(K[I]) + div(K[J]) = yCdiv(I)) + y(div(J )) , so
y
THEOREM 16.7. Assume that a Krull domain.
is a homomorphism. K
is
$ :C (S)
The mapping
d>C [div(I) ]) = [div(K[I])]
a field and -► C(K[SJ)
K[S] defined
is an isomorphism of C CS)
is by
onto
C(K[S ]) . Proof. (3)
Let
y:P(S) -- ► P(K[S])
of Theorem 16.6.
div(XyK[S])
Since
<J>.
injective and surjective. is principal and
K [Iv ]
part
y(div(y + S)) * div(K[y + S]) =
, it follows that
duces the homomorphism
be defined as in
p(P(S)) s P(K[S]) , so
We show that If [div(I)]
<J>
is
in-
both
e ker <j> ,then
contains monomials.
p
Therefore
K [Iv ]
220 K [Iv ]
is generated by
Xa
for some
K [Iv ] = XaK[S] = K[a + S] so
[div(I)] = 0
and
a e G .
implies, however, that
plies that
K[G] = K[S]N , where K[G]
is a Krull
domain,
C(K[G]) = {0}
shows that
is generated by
minimal prime of
K[S]
Qa
{[div(Qa )] | Qa
Ia of
<J>
a ,
is a
Consulting Corollary
is of the form
[div(K[Ia ]) ] = ()>([div(Ia ) ])for each , and hence
is surjective.
Therefore Theorem 16.1
N) .
an appropriate proper prime ideal
C(K[S])
4>
and then Theorems 15.1
.
that meets
15.9, we conclude that each
range 4> 2
= a+S,
N = {Xs | s e S} ,im
and 14.15 show that C(K[S])
I
is injective.
To complete the proof, we show that The equality
The equality
K [Ia ]
S .
for
Since
it follows that
is surjective.
This com
pletes the proof of Theorem 16.7. A combination of Theorems 16.3 and 16.7 yields the following corollary. COROLLARY 16.8. domain, then
C(D[S]) ~ C(D)
Assume that
S
and assume that of
S .
Let
If the monoid domain
H = {0}
{W } » a aeA
7
G
that are essential for
Pa
on
be the center of
{e } » a ae A
wa
associated with S .Theorem
F =
’ ™ ^ ere
family
a:G -- ► F
S , let
w^
Wa , and let
15.8 shows that
be the weak direct sum of the Za = Z
^or each
be the canonical free basis for
the mapping
G
be the family of rank-one discrete
G
Let
© C(S) .
is the set of invertible elements
be a normed valuation on
.
a Krull
is a Krull monoid with quotient group
valuation monoids on
S = nae^Wa
D[S] is
defined by
a » anc* let F .
Since
a (x) = Eawa (x)ea
H = {0} , is an
221 imbedding of
G
in F . Part
that
=S
in this case, and
G n F+
(3)
ofTheorem 15.2 it is an
S
shows
of this form
to which Theorem 16.3 reduces the problem of determining the divisor class group of a Krull monoid domain. THEOREM 16.9.
Let the notation and hypothesis be as in
the preceding paragraph. (1)
*S a ^ree basis for
(2)
Let
set of {wa } value on Then
y € G\{0}
and let
i
div(y + S) = divCE^w i(y) C(S)
- F/G
each of
(1)
and
Krull domain
(3)
{div(K[Pa ])} (divfP^)}
*
if
w^(y) < 0 .
(2) , In proving K[S]
K , and we transfer properties of the
K[S]
That
S .
(2) , we consider the monoid domain
over a field
from part
on
l
We remark that in the statement of
means -w^(y)(S:P^)
(1):
w.
.
Wi(y)Pi
S
be the finite sub
P. for the center of
7
Proof.
of
.
consisting of those valuations with nonzero
y .Write
(3)
P(S)
to
S .
(div(Pa )}
is a free subset of
P(S)
follows
of Theorem 16.6 and the fact that is a free subset of p(K[S])
generates
P(S)
To
prove that
, it suffices to show that
belongs to the subgroup generated by tegral divisorial ideal
.
I * S
of
(div(Pa)} S .
Part
div(I)
for each in (2)
of
Theorem 16.6 shows that
K [I] is aproper divisorial ideal
of
15.9shows
K[S]
primes of (Qj>
T
, and Corollary K[S]
containing
*s t*10 subset of
that contain
I .
Hence
{ pa }
K [I ]
that the set is
of minimal
(K[Q^.]}?=1 , where
consisting of elements
div(K[I])
P^
belongs to the subgroup
222 of
P(K[S])
generated by
{div(K[Q^])}™ , and since each of
these elements is in the range of the mapping 16.6, it follows that P(S)
generated by (2):
(2) , it suffices to consider the case
K[y + S] = X^K[S]
15.7.
(3):
Let
w^
div(y + S)
jective. ^lm iei
(1)
as in the proof of Theorem
a(G)
According to parts
i
and
But this is precisely the condition that kerT
= a(G)
as asserted, and
Thus, if
(DtkP^^,
is
(1) , however, this
y e G\{0}
w a (y) = 0
.
=
if and only if
occurs if and only if there exists for each
t
F , then
and
is sur
t
is the kernel of
(2)
F
determined by
implies that
Znu [div(P^)] = [divCEm^^)] = 0
w^(y) = nu
is the valuation on
t:F -- * C(S)
a nonzero element of
principal.
is deter
canonical free basis for
Part
We show that
w*
k^
= div(I^w^(y) P^), as asserted.
and consider the homomorphism T (ea) = [div ( Pa) ] .
Moreover, since
the integer
, where
^e a ^aeA
div( y + S) =
in this case.
associated with
Therefore
shows that there exist
such that
is principal,
w*(Xy ) = w^(y)
q.f.(K[S])
(1)
k^,...,k
Elkidiv(Pi) = div(EikiPi)
mined as
belongs to the subgroup of
The proof of
positive integers
of Theorem
(divCQj)}™ .
To prove
y e S .
where
div(I)
y
for
such that a J {l,...,n} .
Em^e^ = o(y)
C(S) ^ F/a(G)
c* F/G .
. Thus This
completes the proof of Theorem 16.9. Since Theorem 16.9 provides an alternate description of C(S)
in the case of a Krull monoid
vertible element,
S
with no nonzero in
its combination with Theorems 16.3 and 16.7
yields the following result.
223 COROLLARY 16.10. Let
H
press let G
Assume that
D[S]
is a Krull domain.
be the group of invertible elements of S
as
H © T .
Let
G
S , and e x
be the quotient group of
{wa^aeA
T ,
rank-one discrete valuations on
that are essential for
direct sum of groups imbedding of
G
in
T , let
F =
Za = Z , and let F .
Then
be the weak a
C(D[S])
be the canonical C(D) © (F/a(G)) .
We conclude the section with a result concerning Krull monoids that follows from Theorem 16.9; the statement of Theorem 16.11 is an expected result. THEOREM 16.11. group
^ QL^0Lef^
G , let
valuation monoids on Pa
Let
S
be a Krull monoid with quotient
be the family of rank-one discrete
G
that
be the center of Wa
on
of minimal prime ideals of Proof.
Clearly each
S , and
let
S .Then
is
set
^pa ^aeA
t*10
S . Pa
noinclusion relation between Thus, to prove
are essential for
is prime in
S , and there is
Pa and
for
Pg
a * 3 .
(16.11), it's enough to show that each prime
ideal
P
of
S
contains some
where
H
is the group of invertible elements of
definition, each
Pa
Pa .
is contained in
straightforward to show that
Express
S
as S .
H © T , By
T , and in fact, it's
is the family of centers
of rank-one discrete valuation monoids on the quotient group of
T
P
by
that
that are essential for
T , a Krull monoid.
Replacing
P n T , we therefore assume without loss of generality S = T .
Choose
x e P .
Part
(1)
implies that there exists a finite subset and positive integers
of Theorem 16.9 ^P^}^
such that
of
(Pa )
224 x + S = hence
. P
Therefore
contains some
P ^ x + S 2 ^k^P^
, and
P. . 1
Section 16 Remarks Detailed treatments of the material cited in the intro duction to this section concerning the
v—operation on a
Krull domain can be found in [20; Ch. VII, Sect. Ch.
I and II], and [51; Sect.
34, 44].
1],
[44;
In particular,
is a special case of Theorem 7.1 of [44].
(16.1)
Other references
for the divisor class group of a Krull monoid domain are [95],
[7],
[28], and [21].
More specifically, part
(16.2) is Proposition 5.3 of [95], part is part
(1)
of
of
of (16.3)
of Proposition 7.3 of [7], and Corollaries 16.8
and 16.10 appear in [28]. culations of
(1)
(3)
CCD[S]) , see
For some specific examples of cal [6— 8 ], [28], and [21].
\
CHAPTER IV RING-THEORETIC PROPERTIES OF MONOID RINGS
Chapter IV continues the main theme of Chapter III — that of determining,
for various ring— theoretic properties
necessary and sufficient conditions on a unitary ring a monoid
S
in order that
R[S]
R
should have property
E , and
E .
The main difference in the two chapters is that in Chapter IV, R[S]
is not normally assumed to be a domain; hence
contain zero divisors and
S
R
may
is not uniformly taken to be
torsion— free and cancellative.
In Section 17,
E
is the pr o
perty of being (von Neumann) regular, and certain chain conditions,
including
d.c.c.
, are considered in Section 20.
It is trivial to determine conditions under which a monoid d o main is regular or Artinian (that is, a field). hand, the properties
On the other
E considered in Sections 18 and 19, in
cluding the conditions of being a Priifer ring or an arithmet ical ring, are closely related to properties considered in Section 13 of Chapter III.
One new wrinkle arises in the case
of rings with zero divisors, however.
While a unitary inte
gral domain is arithmetical if and only if it is a Priifer domain, in Section 19 we show that if then
R[S]
S
is not torsion— free,
may be arithmetical, but not a Priifer ring.
Similar situations exist in regard to the properties Bezout ring-arithmetical ring and the properties of being a princi pal ideal ring or a
ZPI— ring. 225
226
§17.
Monoid Rings as von Neumann Regular Rings
Throughout this section we use the term regular ring for a ring that is regular in the sense of von Neumann. inition, for that each x e R .
a ring
a e R
R
The def
that need not be commutative, is
isexpressible in the form
For
R
commutative,
R
pal ideal of
R
is idempotent iff
axa
for
some
is regular iff each princi R
is a zero— dimensional
reduced ring. We assume throughout the section that (commutative) ring and that
S
is a monoid.
results are directed toward a proof that mensional if and only if periodic.
Since
inequality
R
dim R £ dim
torsion— free rank = a
is a unitary Our first
R [S ]
R [S ]
Assume that a , then
is zero— d i
is zero— dimensional and
is a homomorphic image of
THEOREM 17.1.
|A|
R
R
R[S]
S
is
,the
is always satisfied. G
is a group.
If
G
dimR[G] = d i m R [ { X . K A ] A AeA
has , where
.
Proof.
First, a comment on the statement of the theorem
seems appropriate.
We do not distinguish among different
infinite cardinalities in dealing with Krull dimension, so if
a
is infinite, the equality in the statement of (17.1)
is interpreted as meaning that Let
R[G]
be a maximal free subset of
H =
he the subgroup of
Then
G/H
is a torsion group
so
R[H]
and
dimR[G] * dimR[H]
.
rank of
is infinite-dimensional.
H
ality that
is
G
generated by R [G ]
G , and let {y^)
•
is integral over
Moreover,
the torsion— free
|I| = a , so we assume without loss of gener
G = H .
In that case,
R [G ]
is isomorphic to
227
R [( \ >A eA *( \ R[(X
A
*
T ^e latter ring is integral over
+ X ^} ] , for each of
X
A
monic polynomial
Y 2 - (X
and
A
X *
+ X *)Y + 1
A
is a root of the
A
over
A
R[{X
+ X*}]
A
.
A
A straightforward degree argument shows, however, that {X
+ X *}
A
is algebraically independent over
A
quently,
dimR[G] = dim R [{ X ^
COROLLARY 17.2.
If
is periodic, where Proof.
~
^
dim R £dimR[S/~]
since
R[S]
dimR[S/~] = n
group of R[G]
S/~ .
Then
and
S/~
Proof.
dimR
.
Let
R[S/~]
S .
G
.
be the quotient since
The equality
Consequently,
G
of
dimR=
G
has
is a torsion group
is periodic.
dimR[S] = k
and
is a homomorphic image
dim R £ dimR[G] £ dimR[S/~]
0 .
THEOREM 17.3. Then
S/~
*
k 2) \
is a periodic element of
meets the semigroup
< s> of
.
S .
S
Assume
228 that
I n <s> = <J> .
ideal U
J
of
S
By Theorem 11.8, there exists a prime
containing
be the submonoid
R[ J]
S\J
is an ideal of
quently, so that
I
of
R[S]
such that
S .
, and
R [ U ] ^ R[S]/R[J]
Then R[U]
J n <s> = <J> . R[S] = R[U]
n R[J]
k = dim R < dimR[U] < dimR[S] = k , and
U/~
is periodic.
k^s + u = k 2s + u k2
Since
u e U .
contradicts the fact that periodic and
S
and
R [S ]
R[U]
, we see
But then
k^
and
u e J n U , which Consequently,
s
is
is periodic, as asserted.
R is reduced,S
pis regular on
p— torsion— free.
Conse
s e <s> c U , it follows that
J n U = <J> .
Theorem 9.17 shows that only if
.
dimR[U] =? k .
for distinct positive integers
and some element
+ R [J ] ,
is a homomorphic image of
Applying Corollary 17.2 to the monoid ring that
= (0)
Let
R
R[S]
is a reduced ring if and
is free of asymptotic
for each
p
such that
torsion, S
is not
This result and Theorem 17.3 enable us to
determine the monoid rings that are regular. THEOREM 17.4. a monoid.
Let
The ring
R
R[S]
be a unitary ring and let
S
be
is regular if and only if the
following four conditions are satisfied.
S
(1)
R
is regular.
(2)
S
is free of asymptotic torsion.
(3)
p
is regular on
is not (4)
for each prime
p
such that
p— torsion— free. S
Proof. R
R
is periodic. If
R[S]
is regular,
is a homomorphic image of
and Theorem 17.3 implies that
R[S] S
then .
R
Thus
is regular since dimR=dimR[S] =0 ,
is periodic.
Moreover,
229
R[S]
reduced implies that
Therefore conditions regular.
(2)
(1)— (4)
and
(3)
are satisfied.
are satisfied if
invertible, condition
(3)
the condition that
is a unit of
such that
S
p
is not
tic of
for each
and only if
R
R
and if
p
A e A , then
p
p J {p } . A A €A
>
for each prime
Thus,
If
{M_}_ , A AeA
A
is a unit of
R
if
stated in terms of
A , where
p^ * 0 , then condition
in particular,
S
is
"0— torsion— free " is in-
(3)
Note that if
implies condition
this is the case if
R
(2)
;
has nonzero character
To obtain other equivalent forms of the conditions in
Theorem 17.4, we consider some consequences of nation of conditions THEOREM 17.5.
(2)
and
element
Moreover,
t
of
T .
(4)
if
is periodic, then
T
prime
p if and only if
p
As in Section 2, let
and index, respectively, of
nition of
If
r > 1 , then r .
k(m + r - l)t
But for since
is free of asymp
T
divides the order of no element of
r = 1 .
T
is a group for each periodic
p— torsion— free for a given
Proof.
the combi
of that result.
If the semigroup
totic torsion, then
is
is the
is the characteris-
terpreted as "free of asymptotic torsion".
istic.
p
can be replaced by the condition that
p — torsion— free for each
some
regular ring are
of Theorem 17.4 is equivalent to
p— torsion— free.
set of maximal ideals of
A A €A
is
The converse follows by similar reasoning.
Since regular elements of a von Neumann
{Pi}-* a
R[S]
t .
T . m
and
r
be the period
It suffices to prove that
(r - l)t *
(m + r - l)t
by defi
k > 2 , we have k(r - l)t =
k(r - 1) = k(m + r - 1) (modm)
and
230 since
k(r — 1) ^ r .
This contradiction to the fact that
is free of asymptotic torsion shows that
T
is a group, as
asserted. Since any two elements of a it follows that if no nontrivial ible by
T
is
p
Conversely, if
T
is not
contains
generate the same subgroup of ments of the subgroup G since
and b
such that
c , then
T ; hence
G = d
p— torsion— free, then
b,c e T
does not divide the order of
b + d * e
T
p— group, and hence no element of order divis
p .
identity of
p—equivalent,
p— tors ion— free, then
there exist distinct elements If
p— group are
of
T .
b If
is the inverse of
c
and
and e
c
pc are e l e
is the
c
in
* c , but p(b + d) = e .
divides the order of an element of
pb = pc .
G , then Therefore, p
S .
In view of Theorem 17.5 and the remarks following Theorem 17.4, we can state a variety of conditions on S
that are equivalent to the condition that
R[S]
R
and
is reg
ular . COROLLARY 17.6. is the set char(R/M^) = p^
Assume that
R
of maximal
for each
A e A
is regular,
that
ideals of .
R ,and that
Assume that
odic and free of asymptotic torsion.
S
is peri
The following conditions
are equivalent. (1) is not
p
R for each prime
p
such that
p— torsion— free.
(2) the order (3)
is regular on
p
is a unit of
of an element of Nop^
divides the
R for each
prime p
that
divides
S. order ofan element of
S .
S
231 If the semigroup
T
is periodic and free of asymptotic
torsion, then Theorem 17.5 implies that periodic subgroups.
T
is a union of
We embark on a brief consideration of
this and some related conditions in a semigroup
T
.
Thus,
the material in the rest of the section is primarily con cerned with some of the theory of commutative semigroups that was not covered in Chapter 1.
In particular, we abandon for
the rest of the section the restriction to consideration of monoids that has been the rule for several sections.
We use
the term semilattice to describe a semigroup, each of whose elements is idempotent.
The results concerning the
Archimedean decomposition of a semigroup established in Theorems 17.9 and 17.10 will subsequently be used in Section 23. THEOREM 17.7. of a family
Assume that the semigroup
^ a ^a€A
subgroups of
semilattice of idempotents of u{Ga | e
subgroup of and
H
e
G,
periodic, then each If
for Hg
idempotent of e^ € t + T . t + e = t (e d
T
with
To prove
and e = t + y
+t)+y = e + Ol
E
be the
e E , let Each
Hg
Hfi = is a
Moreover,
if each
G , G„
is
a
is also periodic.
t e T , then
t + x = e . a
is the union
a partition of
e,f e E .
be the identity element of such that
Ga > .
the family
+ Hr E H ~ f e+f
Proof.
T . Let
T , and for e
is the identity element for
T
t e Ga
Ga .
for some
Then there exists
We observe that
e a
the properties that t this, assume that for
(t+y)®e CX
a .
some
y +
€T
e
Let
ea
x e Ga
is the unique n + e^ = t
e
eE
.
Then
and issuch that
e = t+ y
= (x +
t)+
= ol
e = x + (t +
232 Thus, and
He
may be alternately described as
e e t + T } .
Since each
potent and since T= uG^ is a partition of T , take
a,b e
closed.
G.
He .
and e = e + e e
, it
G
a
contains a unique idemn
follows
To prove that
Then
(a+b)
(a + T) + (b + T)
It is clear that
{te T | t + e = t
e
easily that Hg
is a subgroup of
+ e = a +
(b + e) = a +
c G
(1) G
.
Thus, assume that
T =
into periodic subgroups
is periodic.
asymptotically equivalent. a
torsion.
If
Assume that
s e G
a and
and
s,t e T
t e G
, c G„ , <s> n a 3 Choose relatively prime positive integers n and
3
G^ . are
, then * d>
m such
.
233 that
ns = nt
tegers
x
and
and
ms = mt
y .
, and write
Since
1 = nx + my
is a group, we have
nxs + mys = xnt + ymt = t .
Therefore
T
for in s =
is free of asymp
totic torsion. Assume that
T
is a periodic semigroup that is free of
asymptotic torsion. to
T , where
is the family of cyclic subgroups of
T , then elements of
We remark that if Theorem 17.7 is applied
T
a ,b e T
belong to the same subgroup
if and only if the identity element for
same as the identity element for
He
is the
< b> .
Theorem 17.7 is a special case of a more general decom position theorem for commutative semigroups that we proceed to establish.
To partially motivate the development, we note
that in Theorem 17.7, the set
E
of idempotents of
semilattice and that the partition with the operation on for all
e,f e E .
E
T = ueegHe
in the sense that
U = u
U + U c U a p a +p
is a
is compatible
Hg +
c ^e + f
In general, we say that a semigroup
a semilattice of subsemigroups semilattice,
T
AU aeA a
for all
is a
{ua |a ^
e A} if A
is
partition of U ,
a ,3 e A .
U
is
a and
The decomposition theorem
referred to above states that each commutative semigroup
T
admits a unique decomposition as a semilattice of Archimedean subsemigroups, where a semigroup
U
rad(x + U) = rad(y + U)
x,y e U .
for all
is Archimedean if (The terminology
Archimedean is suggested by the theory of ordered groups, for such a group is Archimedean, as customarily defined,
if and
only if its semigroup of positive elements is Archimedean as defined above.)
To obtain the
decomposition theorem,we
introduce a new congruence on a semigroup
T
.
234 THEOREM 17.9. on
T
by
s p t
(1)
p
U = T/p
if
is a congruence on
T/y
(2)
Let
under
T , define a relation
rad(s + T) = rad(t
is a semilattice.
such that
T
For a semigroup
.
T , andthe factor semigroup
If
y
is a semilattice, ^ u ^u€u
+ T)
p
is any congruence on then
T
y > p .
be the set of equivalence classes of
p , indexed by the elements of under the
U
canonical
in such a way
that
Tu maps to
u
map f:T
Then
T = u ueuTu
is the unique representation of
-► U
.
T as
a
semilattice of Archimedean semigroups. Proof. is clear.
(1):
To prove that
operation
on
ns e t + T
That
p
n e
t + x + T .
s p t
Z+ , so
and that
n(s +
s + x e
Consequently,
is an equivalence relation on
is compatible with the semigroup
T , assume that
forsome
p
radft + x + T)
.
radft +
x + T)
c
rad(s + x + T)
so that
We have
t p
2t
Assume
that T/y
such that s + y
,so
and
and
some
(s +
t)y (2s + y ) y (s + y)y
t.
To prove Tu of
y>
p.
is a semilattice. s,t
e
ms = t + x
t y nt y(s + y) s and
t and
.
T and
are nt =
x,y € T . Thus
t)y(2t + x)y(t + x)y
s y
, and hence
(s + x)p (t +x)
T/p
(s +
Therefore
class
+ x)
Z+
s
Similarly,
This implies that
for some m,n €
Then
x) e t +nx + T
is a semilattice and that
s p t .
s y ms y(t
te T
e T .
rad(t + x + T)
rad(s + x + T) e
for each
x
, so that
This establishes
(1)
(2) , we first show that each equivalence T under p
is an Archimedean subsemigroup
.
2 35 of
T .
Thus, take
2a y(a + b) Therefore
.
Moreover,
Since
a y 2a
a + b e Ty ,
To show that x,y e T
a,b e Tu .
and
a y b , then
and hence
Tu
ma
= b + x
.
is a subsemigroup of
Tuis Archimedean, choose
such that
a y(a + b)
and
m,n
T .
e Z+
and
nb
=
a +y .
Then
(m + l)a = b + a
+ x ,where
a e rad(a
+
x +T) £ r a d ( a + T)
It follows that
rad(a + T) = rad(a + x
+
T), and
a + x e Tu .
The equation
implies that
a
+ l)a = b + (a + x)
belongs to the radical of
by the same argument, Therefore
(m
b
belongs to
Archimedean.
Clearly 7
T = u fTT ue U u
rad(a + Ty )
only show that
f(a + b) = u +
v
Wc
a,b e W c
is Archimedean,
it is
°f
T
= v , and
, b e W, , x e W_ , and d e ’
p b
clear that a p b
if
and that
c,d,e,f e C
y e VI ~ . J f
Then
and
b + x e W (j+e , so
c = d + e and hence
d +
e = c .Considering the equation nb = a
clude by similar reasoning that and
a
and b
it
a
nb = a + y .
c
as a semilattice
if and only if
b + x
Choose
T = uUeyTu ,
a,b e T ,
. For
Conversely, assume that a p b
a e w
T , and
for each a e Ty, be Ty .
a,b c W c . and
is
Tu + T v - Tu+v * we neec*
T = uce^ c
of Archimedean subsemigroups
Wc
Tu .
.
take any representation
Since
in
Ty
To prove uniqueness of the representation
suffices to show that
Tu ;
r
This is clear, however, since f(a) = u , f(b) f (a + b) = f(a) + f(b)
in
is a partition of
the indexing is such that to show
hence
then
b + Tu
r a d ( a + T ) = rad(b + Ty ) , so each
.
c+ d = d .
belong to the same
This completes the proof of Theorem 17.9.
set
ma
.
=
such that
ma e W
me
= W
c
c + d = 2d + e = + y , we con Hence Wc
c = d ,
of the partition.
236 With the notation as in Theorem 17.9, the semigroups are called the Archimedean components of
T.
is Archimedean,
of G .
for
G
is the only ideal
Any
Tu
group
G
Thus,
uniqueness of the Archimedean components implies that the subgroups
H0
of
T
in the statement of Theorem 17.7 are
the Archimedean components of
T .
It is in this sense that
Theorem 17.7 is a special case of (17.9).
For
T
free of
asymptotic torsion, Corollary 17.11 shows that a given Archimedean component
U
of
T
is a group if and only if
U
contains an idempotent. THEOREM 17.10.
The Archimedean components of
cancellative if and only if Proof.
T
If
a,b e T
then it is clear that
is free of asymptotic torsion.
such that
radfa + T) = rad(b
na = nb
and
na + a = nb + b = na + b
.
follows that
T
a = b , so
a,b,c
ponent
such
U of
T
such that both and
na
and
0 < i < k , then we have
U
, of
so
a
T
and
.
b
Choose Then
U is cancellative,
it
T
is free of asymptotic
are elements of an Archimedean com that a + c = b nb
; hence
na + b + b = na + 2b .
are
is free of asymptotic torsion.
+ c . Choose
belong to
nb = c + y . Then
b + c + x = n a + b
+T)
(n + l)a = (n + l)b .
Since
Conversely, assume that torsion and that
T
are asymptotically equivalent,
belong to the same Archimedean component
c + x
are
Assume that the Archimedean components of
cancellative.
n e Z+
T
c + U , say
ne
Z+
na =
(n+ l ) a = a + c + x =
(n + 2)a = (n + l)a + b =
By induction,
it follows that if
(n + k)a = (n + i)a +(k — i)b .Similarly,
(n + k)b = (n + i)b + (k - i)b
for 0 < i
< k.
In
237 particular, 2na = na + nb = Therefore a = b .
a
2nb
and
b
and
are asymptotically equivalent, and
It follows that
COROLLARY 17.11. torsion and that Then
U
U
U
is cancellative, as asserted.
Assume that
T
element for rad(a + U) vertible in
U
T .
contains an idempotent.
We need only show that the existence of an idem-
e e U
implies that
is free of asymptotic
is an Archimedean component of
is a group if and only if
Proof. potent
(2n + l)a = (n + l)a + nb = (2n + l)b .
implies that U U .
is a group.
Theorem 17.10
is cancellative, and hence If
a e U , then
implies that U .
U
e
is an identity
e e rad(e + u) =
e e a + U , and hence
Therefore
U
T
is in
is a group.
To conclude this section, we show asymptotic torsion, then
a
that if
T is
free of
can be imbedded in a semigroup
that is a semilattice of subgroups. THEOREM 17.12. torsion and let
Assume that
T = u ,tT ueU u
T
is free of asymptotic
be the representation of F
a semilattice of its Archimedean components. quotient group of
Ty
for each
imbedded ^in a semigroup lattice of subgroups Proof.
such that
Wu , where
.
a,b e T y
a + d = b + c .
Then
Gu
Wu ^ Gu
for each Gu
[a,b] = [c,d]
Since the sets
Ty
be the
a semi u e U .
in the form
that is, as equivalence classes
and where
as
T can be
W = uu£yWu is
We represent the elements of
used in Section 1 — where
W
u e U
Let
T
[a,b]
,
if and only if
are disjoint, the groups
238 Gu
are also disjoint.
operation [c,d] of
+
on
W
e Gv , then
Gu+y .
[a'jb1]
Let
W = UU 6JJGU •
as follows. [a,b] + [c,d]
If
We define an e Gu
[a,b]
is the element
This operation is well-defined,
and
[c,d] = [c',d!] , then
c + d f = c1 + d
so
for if
on
and
[a + c,b + d]
W .
T . The operation on
Gufor each Since
W , it
G
u
u
c G v - u+v
and
= [a’+ c f,b! +
+ d ’] .
is associ
agrees with the operation each
Gu
is a subgroup of
by the very definition of J
a semilattice of subgroups.
+
J
follows that W = uue^Gu
isa representation
of
on W
as
Finally, we show that the
natural inclusion mapping of identified with
W
e U , and hence
+ G
[a,b] =
a + b* = a 1 + b
Associativity of the operation follows because on
[a + c , b + d ]
that (a + c) + (b' + d ’) =
(a* + c ’) + (b + d)
ative
and
T
into
W , where
a e Tu
is
[2a,a] e Gu , is a semigroup homomorphism.
This is true since [2a,a] + Hence, we can take
[2b,b] = [2(a +
Wu = Gu for each
b),a + b]
u e U in the
. state
ment of Theorem 17.12. Section 17 Remarks The problem of determining conditions under which a semigroup ring
T[U]
cases other than where U
is regular has been investigated in T
is a commutative unitary ring and
is a commutative monoid.
For example, Gilmer and Teply
show in [61] that the statement of Theorem 17.4 remains valid if the hypothesis that
R
is unitary is dropped;
reason we have chosen to state condition terms of Connell
p
being regular on
R .
If
(3) U
this is one
of (17.4) in
is a group, then
[36], using previous work of several authors, shows
239 that
T[U]
is regular if and only if
T
is regular,
locally finite, and the order of each element of unit of
T ; here neither
mutative.
T
nor
Finally, Weissglass
U
tive and
U
T
nor
is
is a
is required to be com
[131] has considered the
problem of determining conditions under which regular, where neither
U
U
U
T[U]
is
is required to be commuta
is not necessarily a monoid.
While Boolean rings form a subclass of the class of regular rings, the problem of determining the semigroup rings T[U]
that are Boolean can be solved using only first prin
ciples. T
The result is that
is Boolean and
U
T[U]
is Boolean if and only if
is a semilattice.
conditions follows from the equality
Necessity of these
(tXu )2 = t2X 2u
fact that a Boolean ring is commutative;
and the
sufficiency follows
easily from the given conditions and the observation that T [U]
has characteristic
2
if
T
is Boolean.
The material in Section 17 concerning the representation of a commutative semigroup as a semilattice of its Archimedean components has been applied in work on questions concerning semigroup rings; [128-129], and [30].
see, for example,
[132],
240 §18.
Monoid Rings as Priifer Rings
In Section 13, conditions on a monoid domain
D[S]
have
been given in order that it should be a Priifer domain, a Bezout domain, a Dedekind domain, or a
PID .
In this section
we consider certain analogues of these classes of domains for rings with zero divisors. named are, respectively,
The analogues of the classes just the classes of Priifer rings, Bezout
rings, general ZPI— rings, and principal ideal rings
(PIR’s) .
At the same time, we consider here the classes of arithmetical rings and multiplication rings; an arithemetical integral d o main is the same as a Priifer domain and the concepts of
multi
plication domain and Dedekind domain agree, but these equivalences do not carry over to the case of rings.
Each of
the classes named is treated to some degree in [51], except for the class of Priifer rings.
On the other hand, the con
cepts may be unfamiliar to some readers, so we develop in this section enough of the theory to facilitate our treatment of the monoid ring characterization problems.
More detailed
references are given in the Section 18 Remarks. Throughout the section, S
R
denotes a unitary ring and
denotes a torsion— free cancellative monoid.
We say that
R
is a Priifer ring if each finitely generated regular ideal of R
is invertible.
the property that
An invertible ideal An = (a^,...,a£)
it is clear that a Priifer ring
R
A = (a^,...,a^)
for each
n e Z+ .
has Thus,
satisfies the following
condition. (18.1) regular, then
If
a,b e R 2
and if at least one of 2
ab e (a ,b ) .
a
or
b
is
241 For the sake of brevity, we use the this section for a unitary ring (18.1).
ad hoc term
P— ring in
R satisfying condition
Theorem 18.9 shows that the monoid
ring
R [S ]
is a
Prufer ring if and only if it is a
P— ring. Theorem 18.2
lists three elementary properties of
P— rings.
THEOREM 18.2. system in
Let
(2)
If each
R^ is a
(3)
If R[S]
is
ae £
i e
Z+
Choose
a,b
P— ring, then e RN b 2
a R
(a ,b )
P— ring. is a
P— ring.
a
P-ring.
R is
with
3
regular, then there
is regular, 2
e R with b
such that a,b
R^ .
ab e
Thus
aR^ = aR^ , and
implies
that
€R^ , and 2
regular. b
is
There exists
of necessity regular
2
ab e (a ,b }R^ , which implies that
(a2 ,b2) . (3):
in
RN is
P— ring, then
such that
Then ab e
*
(a2 ,b2)RN = o 2 ,e2) . (2):
in
a
oo
R =
P— ring, then
(1): If a,3
a,b e R
= bRN .
a
an ascending se-
such that
If R
Proof.
is
R
(1)
exist
be a regular multiplicative
R , and assume that
quence of subrings of
3R n
N
R[S]
If
a,b € R
, and hence
with
is a
S
and that
G*
P— ring, then Proof.
R
.
b
is regular
But Theorem 12.2
is contracted from
R[S]
.
In
ab e (a2 ,b2) = ( a 2 ,b2}R[S] n R .
COROLLARY 18.3. of
regular, then
ab e ( a 2 ,b2}R[S]
shows that each ideal of particular,
b
Assume that
G
is the quotient group
is the divisible hull of R[G]
and
R[G*]
are
G .
If
R [S ]
P— rings.
It is immediate from Theorem 18.2 that
R[G]
is
242 a
P— ring.
G* {G
Moreover, the proof of Theorem 13.4 shows that
can be expressed as the union of an ascending sequence i m m =1
each
of subgroups, where °
R [Gm ]
R[G*]
is a
G m
r
is a
= G
for each
m .
Hence
P— ring, so Theorem 18.2 implies that
P— ring.
The next result is a generalization of Theorem 13.3. THEOREM 18.4. then either
R
Proof. such that S
is
If
Assume that
s
S
is not invertible. Xs
not invertible in follows that
S
is idempotent.
of
S
is a
P-ring,
is a
group.
r e R\{0}
Choose
is a regular element of rXs = r2f + X 2sg .
and since
s ^ Supp(X2sg) f^
R[S]
is not a group and choose
rXs e (r2 ,X2s) , say
coefficient
and if
a regular ring or
is cancellative,
Thus
S * {0}
S
.
R
Since
R [S ] . Since
r =
s
is
it
f°r some 2 (r) = (r )
f , and this implies that
We conclude that
.
is cancellative,
Therefore
s e S
is regular if
S
is not
a group. We subsequently show that R
R[S]
is a regular ring in any case.
a treatment of the
a P— ring implies that
The key to
case of the rational
THEOREM 18.5.
If
R[Q]
is a
r e R\(0>
.
this result is
group
P-ring, then
ring R
R[Q] is a
regular ring. Proof.
Choose
1 -X
is regular in
Since
R[Q]
{R[Z/n!
]
R[Q]
, so
Corollary 8.6 shows that r(l - X) e (r2 ,(l - X ) 2) .
is the union of the ascending sequence of subrings,
it follows that
.
243 r (1 - X) e ( r 2 ,(l - X) 2}R[X±1 /m ]
for somem e
change of variable we can assume r(l - Xm ) e (r2 , (1 - X m )2) ideal R[X]
in
A = (r2 ,(l - X m )2}R[X] , and hence
{X1} .
A
Therefore
r(l - Xm ) e A .
Z+ .
By
a
that R t X ^ " 1 ] = R[X]{xi}
is comaximal with
.
The
XR[X]
in
is prime to the multiplicative system A
is contracted from
We next observe
, so
that R [X ]
is
a free
R[Xm ]-module with a free basis consisting of(l,X,...,Xm Thus the ideal R[X]
B = ( r 2 ,(l - X m )2}R[Xm ]
by Theorem 12.2, so
R— automorphism of
R[Xm ]
this mapping we obtain
is contracted from
r(l - Xm ) € B . mapping
*}.
1 - Xm
There exists an to
rXm e { r 2 ,X2m}R[Xm ] .
Xm , and under An easy com-
2
putation then shows that
r € r R , and thus
R is regular,
as asserted. The results up to this point enable us to give equivalent conditions for the group ring H
to be a
R [H ]
P-ring or a Prtifer ring
of a torsion— free group (Theorem 18.8).
But in
order to broaden the statement of that result, we consider briefly the notion definition:
R
(A
n
n
B)
+
(A
is C)
of an arithmetical ring. We recall arithmetical if for all ideals
A n (B + C) = A,B,C
dition is equivalent to distributivity of lattice of ideals of
the
of R +
;
this
over
n
con in the
R , and it is also equivalent to the
validity of the Chinese Remainder Theorem in
R .
It is well
known that a unitary integral domain is arithmetical if and only if it is a Prufer domain.
We call a unitary ring
chained ring (or valuation ring) if the set of ideals of is linearly ordered under inclusion.
R
a R
244 THEOREM 18.6. if
R^
The ring
R
is arithmetical if and only
is a chained ring for each maximal ideal
M
of
R .
A Bezout ring is arithmetical and an arithmetical ring is a Priifer ring. Proof. ideal of
R., M
ideals of section,
Assume that R
R
is arithmetical.
is extended from to
Rx.
R
Since each
and sinceextension of
distributes over both sum and inter-
it follows that
R^
is also arithmetical.
Thus,
we assume without loss of generality that
R
is quasi— local
with maximal ideal
R
is a chained
M , and we prove that
ring.
It suffices to prove that the set of principal ideals
of
is linearly ordered under inclusion.
R
a,b e R
and
assume that b J
(a)
(a) = (a) n (b,a - b) = [(a) n Thus, write
a = u +v , where
v = ra = s(a - b) e (a) n then
a = c "''db e (b)
sb = (s - r)a 1 —c - s
with
.
we conclude that
and let
e^
ideals of chained, n .
.
to
and
If
R ,
c
is a unit of
c e M .
R , and since and that
(a - b ) ] .
ca = db e (a) n (b)
Moreover,
s e M .
implies that
t*le set and
R
u =
Otherwise,
a e (b)
converse, let
We have
(b)] + [(a) n
(a - b)
b { (a)
is a unit of
.
Thus, take
(1 - c - s)a = — sbe(b), R
is chained.
m ax im a l ideals
For
the
of
R
c^
denote extension and contraction of
R xt
for each
Mx
A e A
.
Since
RXjI
its lattice of ideals is distributive under
Thus,
Hence
for ideals
A,B,C
of
R
we have
is +
and
245 A n (B + C) = n, . [A n (B + C)]eXCA = A €A
n [Ae^ n (Be^ + Ce*)]c*= n,[(Ae* n Be*) + (Ae* n C6*)]0* = A
A
n. [(A n B) + (A n C)]eXcA = (A n B) + (A n C)
.
A
This completes the proof that if each localization Assume that
R
R^
Bezout ring T
T
is a chained ring.
M
of
is chained.
Let
elements
ar + b s , a = cx, and
x ^ M
and
is Bezout
or
R
be the maximal ideal of
a,b e T .
. Then
Since
If
(c) = (a,b) , c
If
=
c = crx + csy ,
c * 0 , 1 - rx - sy
y | M .
(b) c (a) .
M
r,s,x,y e T such that
b = cy
c(l - rx - sy) = 0 .
c € (a)
R^
R , and hence to prove that
and choose nonzero elements
T , so
Then
it suffices to prove that a quasi— local
then there exist
of
is arithmetical if and only
is a Bezout ring.
for each maximal ideal is arithmetical,
R
so
is a nonunit
x $ M , for example, then
Thus, a Bezout ring is arithmet
ical . Finally, assume that
R
is arithmetical and let
a finitely generated regular ideal element
b e A ,
and let
of R .
first paragraph of this proof.
A
Since
R^ ^
is principal for each A 6 ^ [( b ) :A 6 ^] .
A e A ,
Because
(b)e A:AeA = [(b):A]e A .
A
and hence
be
Choose a regular
(M } , e , and c
A
A
A
be as
in
is chained, e (b) A =
the A6^
is finitely generated,
Therefore
(b) = nx (b)e XcA = nx (AeA[(b):A]e A)cA = nx (A[(b):A])eAcA = A[(b):A]
Since
(b)
is invertible, this proves that
.
A
is invertible,
2 46 and hence
R
is a Priifer ring.
THEOREM 18.7. nomial ring
If
Proof.If
that
(f,g)
f
R[X] and
for each
g
n , it also follows
It suffices
are nonzero elements of We assume that
deg f .
If
deg f = 0
then we can assume that
f
—
We assume that
(f,g)
be
the leading coefficient of
idempotent
generator of the ideal tR
f
thesis.
et
eg = q*ef + r d e g r < deg ef
of for
.
principal.
and
= n + 1 .
and let R .
e
be an
Then
R =
, and
(f,g) =
The ideal
ef
eR[X]
, the leading
is a unit since
someq,r e eR[XJ
It follows that
et« eR = eR . Hence
with
r = 0
(ef,eg) = (ef,r)
r * 0 , the induction hypothesis implies that principal, and if
Then
is principal by the induction hypo
And as an element of
coefficient
f * fh
deg f
of
eR © (1 - e)R , R[X] = eR[X] © (1 - e)R[Xj
((1 - e)f , (1 - e)g)
and we
is principal if
deg f ^ n , and we consider the case where
.
, then
is idempotent.
g = (1 - f + fg)h .
(ef,eg) © ((1 - e)f , (1 - e)g)
R[X]
that i s , if
h = f + (1 - f)g , for
t
to prove
deg f £ deg g
(f,g) = (h) , where
Let
R [Q ] =
Hence, we prove that the poly
is a Bezout ring.
use induction on —
Bezout ring,then its quotient
R[Z/n!] - R[Z]
is principal.
f e R
, and the
are Bezout rings.
is a Bezout ring.
nomial ring that if
is a
R[Z]
is also a Bezout ring. Moreover, since with
R[Q]
R[Q]
R[X]
R [Z ]
u^R[Z/n!]
is a regular ring, then the poly
R [ X ] , the integral group ring
rational group ring
ring
R
r = 0 , it is clear that
or .
If
(ef,eg) (ef,eg)
Since each of the summands of the ideal
is is
(f,g)
247 in the decomposition (f,g) = (ef,eg) © (fl - e ) f , (1 -e ) g ) is principal,
(f,g)
is also principal.
This completes the
proof. THEOREM 18.8.
Assume that
group with divisible hull
H* .
H
is a nonzero torsion— free
The following conditions are
equivalent. Cl)
R[H] is a
C2)
R[H*]
C3)
R
is
P— ring.
is a
P— ring.
a regular ring and H*
(4)
R[H] is a
C5)
R[H] is arithmetical.
C6)
R[H] is a
Proof.
Bezout ring.
Prufer ring.
Corollary 18.3 shows that
Theorem 18.7 shows that cations 18.6.
(4) -**> (5)
(3)
and
implies
(5) = >
(2)
sum of copies of
implies
(3) .
Cl)
implies
(2) ,
(4) , and the impli
C6)
Since each Prufer ring is a
prove that
=Q .
follow from Theorem
P— ring, we need only
The group
H*
is a direct
Q , and in view of Theorem 18.5, it suffices
to show that the assumption that there is more than one Q— summand leads to a contradiction. H*
=
of
Theorem 18.2 implies that
P— ring.
Q ©
Q © K , where
Hence
R[Q]
K
is a subgroup R[Q © Q]
plies
(3) .
* R[Q][Q]
ofH*
.
Part
is
a
is a regular ring by Theorem 18.5, and
this contradicts Theorem 17.4, for semigroup.
Thus, assume that
We conclude that
Q
is not periodic as a
H* = Q , and hence
(2)
im
(3)
248 We are able to extend Theorem 18.8 to monoid rings with out further ado. THEOREM 18.9. that
S
Assume that
R
is a unitary ring and
is a nonzero torsion— free cancellative monoid.
The
following conditions are equivalent. (1)
R [s ]
is a Bezout ring.
(2)
R [S ]
is arithmetical.
(3)
R[S]
is a Priifer ring.
(4)
R[S]
is a
(5)
R
P— ring.
is a regular ring, and to within isomorphism,
is either a subgroup Proof. prove that
of
In view (4)
assume that
Q
or a Priifer submonoid
(5)
R[S] is a
and
(5)
S.
Hence
R
is regular and
sider
S
as a submonoid of
H* s Q Q .
of
Qq
containing
that
If
assume that
S
is a
P— ring.
We con
is a subgroup of S
Q
is isomorphic to a
1 ; without loss of gene
S = U .To show that
h * 0so that
(X2a ,X^k)
.
R[S]
a > b .
S is
a
Prtifer
a + b , we have
where
Since
X
a
a,b e S .
We
b and X
are
, a P-ring, we have
This implies that
a + b e 2b + S .
than
. Thus,
contains both
Take h = a - b e H n Q Q ,
regular elements of
or
only
bethe quotient R[H]
S
.
it suffices, by Theorem 13.5, to show that
H n Qq c s .
Xa+b e
(1)
by Theorem 18.8.
In the contrary case,
rality we assume monoid,
H
Theorem 18.2 implies that
positive and negative rationals, then
U
implies
P— ring and let
group of
submonoid
Q
of Theorems 18.6 and 18.9, we need
implies
by Theorem 2.9.
of
S
As each a+ b e
either
element of 2b + Sand
a + b
2a + S h =
e 2a + S
isgreater a — b
e S .
249 Thus
H n Q q c s , and (4)implies (5) = >
(1):
If R
, as asserted.
is regular and
Q , then Theorem 18.8shows the other hand, if
(5)
that
R[S]
S
is a subgroup
of
is a Bezout ring.
S is a Prufer submonoid of
On
Q , then
oo
write
S = u
S i “ ^i+1 R[S^]
, where each
^or each
* •
is isomorphic to
Therefore
R[S]
is a cyclic monoid and
Then R[X]
R[S] = u^RCS^]
, where each
, a Bezout ring by Theorem 18.7
is also a Bezout ring, and this completes
the proof of Theroem 18.9. The ring-theoretic analogue of a Dedekind domain is the notion of a general
ZPI— ring, defined as a ring in which
each ideal is a finite product of prime ideals. of
[51]
shows that a unitary ring
ZPI— ring if and only if
R
R
Theorem 39.2
is a general
is a finite direct sum of Dedekind
domains and special primary rings, where a special primary ring is, by definition, a local ideal.
PIR
with nilpotent maximal
(The term special principal ideal ring
sometimes used instead of special primary ring.)
(SPIR)
is
For our
purposes, the equivalence in terms of Dedekind domains and SPIR's
could be taken as the definition of a unitary general
ZPI— ring.
Note in particular that a unitary general
is Noetherian.
An ideal
multiplication ideal if that it contains —
A A
of a ring
T
is said to be a
is a factor of each ideal of
that is, if {AB^lB^
is the set of ideals of
T
is an ideal of
contained in
a multiplication ring if each ideal of ideal.
T
ZPI— ring
A ; the ring T
T} T
is
is a multiplication
It is clear that a regular ideal of a unitary ring is
a multiplication ideal if and only if it is invertible, and hence a unitary integral domain
is a multiplication ring if
250 and only if it is a Dedekind domain.
Moreover, each unitary
multiplication ring is a Priifer ring.
On the other hand, a
regular ring is a multiplication ring (for A g B in a ? regular ring implies A = A = AB ) , and hence a multipli cation ring with zero divisors need not be Noetherian. Theorem 18.10 determines necessary and sufficient conditions in order that general
R[S]
should be a multiplication ring or a
ZPI— ring.
While Theorem 18.10 is a summary result,
we do not repeat our standing hypotheses concerning S
and
in its statement. THEROEM 18.10.
The following conditions are equivalent.
(1)
R [S] is a
PIR .
(2)
R[S] is a general
(3)
R[S] is a multiplication ring.
(4)
R
Proof. PID's
ZPI— ring.
is a finite direct sum of fields and
isomorphic to
of
R
Z
(1) = >
and
(2) = >
or to
(3):
is
ZQ .
(2) :
SPIR's
S
Each
, and
PIR
is a finite direct sum
hence is ageneral
Dedekind domains and
ZPI— ring.
SPIR's
are multi
plication rings, and a finite direct sum of unitary multipli cation rings is again a multiplication ring. general
ZPI— ring is a multiplication ring.
(3) “ > (4): ring.
Therefore each
Then
R[S]
regular ring and submonoid of
Assume that
Q .
is a multiplication
is a Priifer ring, and hence S
is either a subgroup of
We show that
that each proper prime ideal By Corollary 8 .6 ,
R[S]
1 - X
P
R of
R Q
is a or a Priifer
is Noetherian by showing R
is finitely generated.
is not a zero divisor in
R[S]
, so
251 CP,1 - X ) Since
P
is invertible, and hence finitely generated. is the image of
(P,l - X )
tion map, it follows that
P
under the
augmenta
is also finitely generated.
A
Noetherian regular ring is a finite direct sum of fields — say
R = F 1@...©Fn .
F^[S]
Thus
R[S] = F ^ [S ]©...©Fn [S ] , and each
is an integral domain that is a multiplication ring,
hence a Dedekind domain. that either
S = Z
(4) = > fields
(5):
F^
S = Zq .
If
and if
PID , and hence
or
Applying Theorem 13.8, we conclude
R = F 1©...©Fk
S
R[S]
is Z
or
is a direct
sum of
Z q , then each F^[S]
= £* 0 F ± [S]
is a
is a
PIR .
Section 18 Remarks Some detailed references on topics treated in Section 18 are the following. rings
[45],
tion rings
Prufer rings:
[82]; general [106],
[108],
[27],
ZPI-rings [56],
[65];
[107],
[66], [4].
arithmetical
[13]; multiplica Moreover,
[91,
Chapters IX and XJ is a general reference for these topics. We remark that a is a ring
T
ZPI— ring, as opposed to a general
in which each nonzero ideal is uniquely e x
pressible as a finite product of prime ideals unitary, then factors of uniqueness).
ZPI— ring,
T
A unitary ring
(if
T
is
are disregarded in considering T
is a
if it is either a Dedekind domain or a
ZPI— ring if and only SPIR .
The choice
of terminology comes from the German Zerlegung Primideale. Much of the material in Section 18 stems from
[59], but
new proofs have been necessary for most of the results of that paper.
This is because Gilmer and Parker in [59] made
strong use of a result of Griffin in [65] stating that a
252 unitary ring
R
is a Priifer ring if and only if
integrally closed
P— ring.
R
is an
While it is true that a Priifer
ring is an integrally closed
P— ring, the status of the con
verse is in question; Griffin's proof shows only that in an integrally closed
P— ring
(a^,...,an ) , with
R , each ideal of the form
ai > ,,,»an -i
regular, is invertible.
P— ring need not be integrally closed, and Priifer ring.
For example, if
elements and if domain closed.
F
D = F + X K [[X ]]
is a
hence need not be
Kis the Galois
is the prime
A
subfield of
field with K, then the
P— ring, but is not integrally
For details of this example, as well as related r e
sults, see [112],
[27], and [53].
a 4
253 §19. Where
S
Monoid Rings as Arithmetical Rings —
the Case
is not Torsion— Free
Results of Section 18 show that if
R
is unitary and
is torsion— free and cancellative, the monoid ring Priifer ring ring.
iff
it is arithmetical
Moreover,
R [S ]is a general
multiplication ring
iff
it is a
iff
is a
it is a Bezout
ZPI— ring
PIR .
R[S]
S
iff
it is a
An examination of
the proofs in Section 18 suggests that the assumption that is torsion— free is used sparingly,
and that it may be possible
to extend some of the results to the case where cancellative monoid.
S
S
is any
This section is devoted to that end.
The key concept in this development turns out to be that of an arithmetical ring, rather than that of a Priifer ring.
Our
results show that even for finite groups, a group ring that is arithmetical need not be a Bezout ring, and that general a
ZPI— ring does not, in general, imply that
R[G] R[G]
a is
PIR . The first four results of the section are devoted to a
determination of conditions under which a monoid ring is quasi-local
(that is, has a unique maximal ideal) or a chained
ring; these results do not require that the monoid in question is cancellative. mining conditions ways: R[S]
They are related to the problem of deter under which
first, a chained
R[S] is arithmetical in two
ring is arithmetical, and second,
is arithmetical if and only if each localization
(R[S])m
of
R[S]
at a maximal ideal
M
is chained.
In
connection with the latter condition, we remark that for a multiplicative system R^[S]
N
in
R , the rings
R[S]^
and
are canonically isomorphic for any unitary ring
R
and
254 any monoid
S .
Throughout the section,
R
denotes a unitary ring and
S
denotes a cancellative monoid. THEOREM 19.1. monoid ring
R[U]
is a
maximal ideal
M
a unit of
F[U]
is a group.
and
and
Let
H
u
particular,
F[U]
U
Then
implies that
Let
Therefore
R[U]
R
is
Xu
is
Therefore
U
F [U ] ; in
is quasi-local. it follows that
p = char(F)
Thus, choose
b = 1
has characteristic
, where
, so
be a prime that divides
X^ - 1 = b ( X - l ) ^
is monic,
Xu
is contracted from a
F[H]
p— group.
To prove the converse, of
p
1-
Theorem 12.2
let I
the ;
u € U
X? — 1
for some unit
and
this of
is quasi-local.
is the only prime factor of
X^ -1 F
F[H]
Then
Then
is contracted from
F[(u)] ^ F [ X ] / ( X P — 1)
X-l
U .
U .
U . We show that
is a
F [X ] , and therefore Since
u
PIR ,
(1):
and hence is either a
valuation domain. to R
We note that a local arithmetical ringis
R ,
If
P< M , then
P R^
and
P—primary ideal, P
tained in
P
i .
primes Q. < Q. x i
M .
is contained in each
Let
(0) = n i= iQ^
(0)
Thus,
P.
i
9
R
It follows that R - E i © (R/Q^)
If
P^
in
P
is the only
M—primary ideal of R
properly con
be a shortest primary r e
R , where
since a relation P^< P^
a maximal ideal of
local,
It follows that
is
P^-primary for
There are no inclusion relations among distinct P^
.
are proper prime ideals of
is the unique prime ideal of
presentation of each
M
is local and arithmetical, hence a
Noetherian valuation domain.
R , and
or a rank-one discrete
This observation has several applications
asfollows.
with
SPIR
and
P.
j
If
are comaximal for
i
* j since
J
properly contains at most one prime ideal. and
.
would imply that
P^
are also comaximal, and hence is
maximal in
R , then R/Q^
zero-dimensional, and arithmetical, hence a is not maximal, then P^
=
and
SPIR .
R/Q^
is a
is
265 Dedekind domain.
Therefore
R
is a general
ZPI— ring.
In view of Theorem 19.12, equivalent conditions for R[S]
to be a general
ZPI— ring can be easily obtained from
Theorems 19.7 and 19.10. THEOREM 19.13.
Assume that S
is not torsion— free. and let
H
Let
If
S = Z ©
order of
H
H
or
R
R .In
If
if
is a finite group, and Proof.
(1):
R
R[S]
is
Assume that
R[S]
a subgroup of
since
and
S , G,H ,and
that
R is
order of
R[S] S/H
and
rated, so
R[S]
S
= G
R[H]
ZPI—ring.
are regular, Q .
is Noetherian
are finitely generated.
It follows
fields,
R , and
S/H - Z Q
is Noetherian and is Noetherian.
Theorem 19.13 shows that
R [S ]
H is finite,
S/H - Z
Conversely, if R ,S ,and R
ZPI— ring,
R
S - H © Z , and similarly,
(1) , then
general
is a general
the proof of Theorem 19.7 shows that if
in
is a
is Noetherian, then
is a unit of
S - H © Zq .
a
Q or a Priifer submonoid of
a finite direct sum of
H
R [S ]
arithmetical.
and
Moreover,
R [S ]
is a general
shows that H c S , R
is
is finite and the
PIR .
Theorem 19.7 S/H
H
this case,
S is periodic, then
ZPI— ring if and only
is a general
direct sum of fields
S =:Z ^ © H , where
ZPI— ring implies that it is a (2)
R[S]
is a finite
is a unit of
S
G .
S is not periodic, then
ZPI— ring if and only if
S
be the quotient group of
be the torsion subgroup of
(1)
and
G
is nonzero andthat
or
S
Z Q . Finally,
S/H - Z , then
implies H
the
that
are as described is finitely gene
Moreover, part
(2)
of
is a Bezout ring, from which
266 it follows that
R[S]
Statement
(2)
is a
PIR .
follows immediately from (19.10), (19.12),
and the fact that R[S] is Noetherian if and only if Noetherian and
S
R = Z[/— 5, 1/11], G = Z ^
R[G]
ZPI— ring.
need not be a
nonzero and not torsion— free?
shows that a
PIR if it is a general
Under what conditions is
R [S ]
Part
a
(1)
PIR , for
S
periodic.
We proceed to consider the case where
periodic.
If
R[S]
is a
PIR , then
is finitely generated, hence finite.
and each
that
R
R^[S]
is a
PIR
S
for each
where
(1)
R
a field, and
R[S]
is a field, (3)
R
is a
is
PIR
PID’s
and
and
if
is a finite i , then
S
is a
PIR
—
is a
SPIR's ,
R = group such
R[S]
Thus, there are three cases to consider for conditions under which
S
It follows that
is a finite direct sum of
is so decomposed and
is not
is a
R^[S] is a PIR . And conversely,
R^©...@Rn
S
of Theorem 19.13
gives equivalent conditions in the case where
R = R.6...0R 1 n
is
is finitely generated.
The example group ring
R
is a
R
PIR .
in determining
namely, the cases
(2)
R
SPIR
that is not
PID
that is not a field.
The
next three results treat these three cases. THEOREM 19.14. istic F[G]
pand that is
a
only if the Proof.
G
PIR .
is a field of character
is a torsion group.
p—primary component If
F
If p * 0 , then
F[G] is a
19.10 implies that G
Assume that
Gp
F [G]
G^
of
If is a G
satisfies the given conditions,
p * 0 .
then
F[G]
= 0 PIR
, then if and
is cyclic.
PIR , then part
is cyclic if
p
(1)
of Theorem
Conversely, if is arithmetical
267 by Theorem 19.10, and finitely generated
F[G]
is Artinian since it is a
F-module.
Therefore
F[G]
direct sum of zero-dimensional local rings. summands is a local chained ring, hence a also a
19.15.
a field and that is a
PIR
Proof.
Assume that
G
PIR , so
if and only if
If
m
Theorem 19.14, is a
m
is a unit of
R[G]
PIR
R is a
F[G]
is
SPIR that is not
is a finite group of order
Theorem 19.10 shows that
R[G]
Each of these
PIR .
THEOREM
R[G]
is a finite
R[ G]
m .
is a unit of R , then part
Then
R . (2) of
is arithmetical.
As in
is also Artinian, and the proof that
is the same as that given in (19.14) for the
corresponding case. If
m
is not a
maximal ideal M m .
Since
of
Gp *
of Theorem
unit of R , then R
{0}and
THEOREM
p
R^ = R
a field, part
D[ G]
(2)
m
is
the exponent of
Proof.
(1)
arithmetical, and
Assume that
D is a
K .
Let
PID G
that is
dis
be a finite group
The following conditions are equivalent.
(1)
unity over
is not
is a divisor of
PIR .
19.16.
m .
is not
R[G]
tinct from its quotient field of order
belongs to the
and char(R/M) =
19.10shows that
hence is not a
m
is a
PIR .
a unit of G
and
K , then
D ,and if
u
is a primitive
D[u]
is a
(1) = >
some prime divisor
of
m
is
is any divisor kth root
of
of
PID .
(2): If p
k
mis not a
unitof
the characteristic of
D ,then D/M A
268 for some maximal ideal
of
D .
Since
is not a A
field, part
(1)
of Theorem 19.10 shows that
arithmetical in this case. m
is a unit of
D .
If
Therefore k
D[G]
D[G] a
is not
PIR
implies
divides the exponent of
G ,
then the fundamental theorem of finite abelian groups implies that there exists a subgroup cyclic of order D[G]
, it is a
D[u]
k .
Since
PIR .
Thus
is isomorphic to
K
of
G
D[G/K]
such that
G/K
is
is a homomorphic image of
D[X]/(Xk -l)
is a
PIR .
D[X]/(f(X))
, where
minimal polynomial for
u
over
D
(which is equal to the
minimal polynomial for
u
over
K)
is a homomorphic image of
D[X]/(X
\r
f(X)
Since
is the
, it follows that
D[u]
— 1) , and is therefore a
PID . (2) = >
(1):
Since
m
is a unit of
Theorem 19.10 implies that D[G]
is a general
D [G ]is arithmetical.
ZPI— ring by part (2) of
over, Theorem 9.17 implies D[G]
is isomorphic to the
{D [
G
D[G]
.
D[G]/P^
]
, where
{
integral over P. n D = (0)
D
Hence
(19.13).
is reduced.
M or e
Hence
^
is the set of minimal primes of is a
PIR , we show that each D
is integrally closed,
are regular in
D[G]
, and
D[G]
is
D ; hence the lying under theorem implies that for each
D[{Xg + P- | g e G } ] . cyclic, generated by
u
(1) of
direct sum of the rings
PID .The domain
nonzero elements of
D [G ]/P^
that D[G]
To show that D[G] is a
D , part
i .
Therefore
^
Consider first the case where h .
Then
as a ring extension of
is a primitive
D[G]/P^
u = X*1 + P^ D , and
G
generates
um = 1 .
Therefore
kth root of unity for some divisor
m , and the hypothesis of
(2)
implies that
is
D[u]
k
is a
of PID .
269 To summarize,
if
G
is cyclic, w e ’ve shown that
isomorphic to a finite direct sum each of
u^
is a root of unity in a fixed algebraic closure
In the general case, express
G
r > 1 .
Let
H = Gj®...@Gr _^ .
and the exponent of exponents of
H
G
u^
divides
G = G-j©...©Gr ,
Then
G = H © Gr ,
is the least common multiple of the
and
Gr .
By induction we assume that
is a finite direct sum of domains
D[v^]
, where
is a
direct sum of rings
thesis on
D
order of
Gr
D[v^][Gr ] , where
implies that each is a unit of
proof in the case where
D[v^]
is a
D , hence of
G
D[v^]
Wj e L
is
(= order) y—
e L
is
H . Thus the hypo
PID .
.
The
Thus, the
is cyclic implies that
is isomorphic to a finite direct of rings
D[H]
v^ e L
a root of unity whose order divides the exponent of D[G]
L
as a finite direct sum
of cyclic groups of prime-power order; say where
is
D[u^]@...©D[un ] , where
K , and where the (multiplicative) order of
m .
D[G]
D[v^,w^]
D[v^][Gr ] , where
a root of unity whose order divides the exponent of
Gr .
We have
D[v^,Wj]
= D[y^^] ,
where
is a root of unity whose order is the least common
multiple of the orders of the exponent of
G .
D[G] = E • -©D [y - -]
1>J
SPIR
and
Thus each
is a
1J
Since an
v^
w^. , hence a divisor of
D[y^j]
PIR .
is a
PID , and
This completes the proof.
is a local chained ring, Corollary 19.2
and Theorem 19.4 yield the following characterization of monoid rings that are THEOREM 19.17. ring
R[U]
is a
characteristic
SPIR's If
SPIR
p * 0
U
.
is a nonzero monoid, the monoid
if and only if and
U
R
is a cyclic
is a field of p— group.
270 Section 19 Remarks The question of whether each chained ring
R
is the
homomorphic image of a valuation domain has been raised in the literature.
If
R
is Noetherian, an affirmative answer
follows easily from Cohen's structure theorem for complete local rings.
Ohm and Vicknair show in [113] that the question
also has an affirmative answer if
R
is a monoid ring; their
proof is based on Theorem 19.4. A unitary ring
R
is said to be semi-quasi-local if
has only finitely many maximal ideals, and if
R
only if
R
is semilocal and
quasi-local if and only if
(2)
is semilocal
is Noetherian with only finitely many maximal ideals.
It can be shown that a group ring
G
R
R
G
R[G]
is semilocal if and
is finite;
(R;M^,...,M )
is a torsion group, and either
(1)
G
char(R/M^) =...= char(R/Mn ) = p * 0
R[G]
is semi-
is semi-quasi-local, is finite, or and
G/G^
is
finite. The problems of determining conditions under which
R[S]
is a Prufer ring or a Bezout ring seem to be open in the cases not covered in Section 18.
Glastad and Hopkins have
addressed the problem of determining when in [62].
R[S]
is a
Specifically, they consider the case where
PIR R[S]
a zero-dimensional ring of the form R fX j, ...,Xn ]/({X®i(1 -X? i) }" =1) , where each f^
et
and each
is a positive integer; each zero-dimensional monoid ring
that is a
PIR
type described.
is the homomorphic image of a ring of the
is
271 §20.
Chain Conditions in Monoid Rings
Two chain conditions —
the ascending chain condition
and the ascending chain condition for principal ideals — have already been considered in monoid rings, the latter for monoid domains. R[S]
To wit, Theorem 7.7 shows that a monoid ring
is Noetherian if and only if
R
is Noetherian and
S
is finitely generated, and Theorem 14.17 shows that a monoid domain
D[G]
and only if of
G
, where D
G
is a group, satisfies
satisfies
is of type
a.c.c.p.
(0,0,...
) .
a.c.c.p.
if
and each nonzero element
In this section we consider
certain other chain conditions in
R[S]
ically, the first part of the section
and
R[G]
.
Specif
(through Theorem 20.8)
is concerned with the problem of determining conditions under which
R[S]
is Artinian or an
RM— ring.
The remainder of
the section treats the property of being locally Noetherian in unitary group rings. Throughout this section we use the notation unitary ring, ring
R
S
G
for a group.
is Artinian if and only if
R
is both zero-dimen
satisfies both
and hence
for a
for a monoid, and
sional and Noetherian. R[S]
R
S
Thus, a.c.c.
satisfies
if and
a.c.c.
R[S]
is Artinian, then
d.c.c. and
The
on kernel ideals,
d.c.c.
on congruences.
We proceed to show that a monoid with these properties is finite.
The proof,
like that of Theorem 5.10, is not imme
diate, but the case where
S
is a group is an elementary
result. THEOREM 20.1. d.c.c.
The group
G
satisfies both
on subgroups if and only if
G
is finite.
a.c.c.
and
272 Proof.
We need only show that
isfies both chain conditions. since it satisfies
a.c.c.
isfy
d.c.c.
Thus,
, so
direct sum of cyclic groups
G
is finite if it sat G
is finitely generated
G = G1©...@Gn
G^ .
Since
on subgroups, each
Z
is a finite
does not sat
G^ , and hence
G , is
finite. THEOREM 20.2.
Assume that
T
is a cancellative semi
group. (1)
If
T
satisfies
d.c.c.
on ideals, then
T
is a
group. (2)
If
T
gruences, then Proof. sequence
satisfies both a.c.c. T
(1):
Choose
k
of ideals of
such that
for some
(2):
Therefore T
T .
T
I
Since
Therefore I
on
satisfies
is a group.
There exists
can be written as T
is cancellative,
is an identity element of
Each ideal
pjwith respect to
T .
kt + T = (k + l)t + T = ...
u e T .
t + u
has an inverse in
on con
and consider the decreasing
(k + l)t e kt + T = (k + 2)t + T
it follows that
that
t e T
t + T 2 2t + T 2 ...
(k + 2)t + u
d.c.c.
is a finite group.
a positive integer Thus
and
of T ,
T
T
and
t
is a group.
T induces the Rees congruence and
I <J
implies
d.c.c. on ideals, and
Consequently,
T
(1)
Pj
, there exists an infinite strictly ascending sequence
{Qj }* If
A
^xa ^aeA *
j , then { P^.}“
(1)
ro(G)
a maxim a l linearly independent
G , and assume that
infinite subset
rank
is finite for each
Theorem 20.9 shows that
(2) , let
subset of
(G/F)^
G of
r o(G) = n Assume that
such that
and let F
be a free subgroup of
p e ft and let
char(R/M) = p .
homomorphic image of
G , the ring
homomorphic image of
R[G]
Since
M
be
G
of
a maximal ideal
(G/F)^
(R/M)[(G/F)p]
is a is a
, and hence is locally Noetherian.
Applying Theorem 20.10, we conclude that
(G/F)p
is finite.
281
In order to prove the converse of Theorem 20.11, we use two preliminary results.
The first of these belongs to the
theory of abelian groups. THEOREM 20.12. the group
then
Assume that
G and that
p
is a finite subgroup of
is prime.
(1)
Gp is finite if and only ifCG/K)^
(2)
If Gp
(G/H)p (3)
is finite and if
G
Assume that
ro(G) = n of rank
finite.
finite and
Then
G
is P tains a free subgroup
and let
(1):
*
of
G
.
is finite and that
nsuch that
G
(G/F)p
con
is
G = G © H , where H conP of rank n such that (^/F^)p =
Let
<j>
be the natural map from
be the restriction of
*(Gp) £ (G/K)p is finite if
and
(G/K)p
* = Gp n K is finite.
is the set of
K .
is finite, then an element
If
root in
Gp
G hasexactly
form a coset
of G
P
in
pth
|Gp| G .
to
.
is finite.
G to
G/K ,
Then Hence
Gp
On the other hand,
Xm ] =
Pq g P^ c...c P^
i
R^m ^
indeterminates over
is said to be special if
of the chain for each
31] or in
First we introduce some notation and
For a unitary ring
the polynomial
Except
(2)
follows from
(1)
and
(3)
(2) .
Assume that
R^m ^ is finite— dimensional and that
289 Q
is a proper prime ideal of
let
(M.K (1)
ht(Q)
.
A Ae A
.
Let
P
= Q
be the set of maximal ideals of
nR
and
R .
ht(Q) = ht(p(m ))+ ht(Q/P^m ^) ; in particular,
can be realized as the length of a special chain of
primes of (2)
R^m ^
with terminal element
dimR^m ^
can be realized as the length of a special
chain
Pq < P^ dim R[G] ^ dim (R/P)[G]
is a finite— dimensional unitary domain.
Let
for
^or
Let
each
i, and choose
{g.}1? ^ be a finite = s i “ ti * with
€ Qj. ~ ^i-i set of generators
s^ , ti e S
for each
for i .
292 Let
T
(s i }^
be the submonoid of ft.}™ u Supp(f1) u.
u
generated,
G
i .
generated by
.uSupp(ft)
is the quotient
dim D[T] ^ t > dim D[G] for each
.
S
torsion— free of rank
, for
S
over
generates
Since
S
If
expressible in the form
GN
M e Z+
, where
has quotient group R[T]
R[xJ*,...,X**]
pure monomials r
Q—vector space,
If
{tj}^=1 t^
is a is uniquely
e Q
.
It
such that for each
T
u { M t j , and hence
finitely generated over
as a
S , then each
Mtj e Zs1+...+Zsn = Zs^©...©Zs^
we're dealing with in
is the set of non
^i=iajisi > where
follows that there exists
R[T]
N
.
finite set of generators for
subring of
is
is ann— dimensional vector space
contains a basis
is integral over
G
n , we can reinterpret the equality
G^
T
D[T])\CQ^_1 nD[T])
is finitely generated and
zero integers, then
over,
T , and
f^ e (Q^ n
asfollows.
by
is finitely
group of
dim R[G] = dim R[S]
S
T
This establishes Theorem 21.5.
In the case where
Q .
Then
.
j .
Then
R[S]
is the monoid generated
dim R[S] = dim R[T] Zs^©...©Zsn = G .
; more
Thus, what
, to within isomorphism, is a containing
R[X^,...,Xn ]
R[X^,...,Xn ]
that is
by a set
h. = X?l...Xan , where each 1 1 n
aj
is an
integer (not necessarily nonnegative); we note that this use of the term ’’pure monomial" is a extension of that used in Section 15, where the exponents negative.
In this notation,
assertion that Since
a^ were assumed to be n on
(21.4) is equivalent to the
dim RfX*1 ,. . -X*1 ] = dim R f X ^ . . . ,X ,1^,. . .,hk] .
dim R[Z^n ^] = dim R^n ^
by Theorem 17.1, we have
established the following result.
293 THEOREM 21.6.
The equality in Theorem 21.4 is equivalent
to the following statement.
If k
unitary domain and if
D
is a finite— dimensional
is a finite set of pure
monomials in the indeterminates
over
dim D[Xr ...,Xn ] = dim DfXj, . . . ,Xn ,(h. }J]
D , then
.
We turn to the proof of the statement in (21.6).
The
first step in the proof is a result about polynomial rings. THEOREM 21.7. P
Let
D
be a unitary integral domain,
be a proper prime ideal of
integers such that ht p^-k) > m + i
Proof. ht
1 < m < n.
for
By passage to ideal
ht p^m ^ £ m + 1 .
m > ht P = dim D . dim
Part
= n + ht p^n ^ >
case that where
dim
K
If
ht
m
P . Thus,
Since
+ 1 , so part
= m
dim
be
> m+ 1 , then
If
D
is
ht P >m + 1 , then
assume that
(3) of (21.1) implies that n + m + 1 .
is the quotient field of
dim
n
But it is also the
= n + sup{dim D[t^,...,tn ] |
dim D[s^,...,sm ] > m Then
and
D p , we assume that
D , by part
Therefore, there exists an overrring of least
m
itsuffices toprove that
quasi— local with maximal certainly
, and let
m< k £n .
Clearly
> m + 1 .
D
let
(2) of (21.3)
+ 1
D
> (4) of (21.3).
of dimension at
implies that
for some subset
+ sup {dim D[t^, . . . ,tm ] |
K • = K } > 2m + 1 .
= m + ht p^m ^ , it follows that
ht P fm^ > m + 1 . If
D
polymomials
is a unitary integral domain, the set f e
B
of
of unit content forms a multiplicative
294 system in
, and the quotient ring
is denoted by
D(X^,...,Xm ) .
of
D(X1 ,...,Xm )
If
Q
.
is a prime ideal of
QD(X^,...,X ) if
Assume that
containing
exists a unique prime
P
n
J
.
Q
P^m ^
of
J
Domains of the form
J
T
such that
result applies to the domains
Let
D
Assume that
D
, there
Q ,
namely Q =
In particular, a
= p(m )
n
, where each
h^
#
is a
P
of
of part
and
Moreover,
D^m ^
J
is
of Theorem 21.6. (1)
of (21.1).
B be as in the preceding
in
is a finitely
D(X^,...,Xm )
and that each
extends to a prime ideal of J .
proper prime ideal of
PT
D , so the following
J = D^m ^[t^,...,tn ]
generated extension of prime of
P^m ^
D^m ^[(h^}]
Theorem 21.8 is a generalization
paragraph.
B
> satisfy the condition that
for each prime ideal
THEOREM 21.8.
is a subring
D , then there exists
s DfX*1 ,....X*1] s D(X1 .... Xm ) .
prime in
of
= (D^m ^)g .
ht Q = ht Q .
T = D^m ^[(hj}^]
pure monomial in D (m) = T
Thus
lying over
Therefore
of
J
that misses
is a proper prime ideal of
unique prime
(D^m ^)g
and if P = Q
n
D ,
If
Q
then
is a
ht Q =
h t ( Q / p W ) + ht( P(m^) 2 m + h t ( P (m)) .
Proof. then
Observe that since
PJ = P ^ J
P (m) = p(m)JB n J = P ^ J
We establish first the inequality
is prime in J
for each prime htCQ/P^J)
P
s ® •
of
D.
Thus,
to within isomorphism,
(D/p)(m) = D(m)/pOO s j/pMj £ ( D ^ ) b / P ^ ( D ^ ) b , and under this identification, N
(Q/P^m ^J) n (D/P) = (0)
is the set of nonzero elements of
D/P
.If
,it follows that
,
295 k ^
c (J/P^m ^J)N c (k^m ^)g* , where
field of
D/P
and
B*
(3)
is the quotient
is a multiplicative system in
Since each overring of part
k
k^m ^
has dimension at most
of (21.3), we conclude that
k^m ^
.
m . by
ht(Q/P^J)N =
ht ( Q / P ^ J) £ m . The inequality
ht Q £ ht(P^m ^J) + ht(Q/P^m ^J)
clearly holds; we establish the reverse inequality by in duction on
ht
Q . If
or
.
In either case, the result holds.
P = (0)
ht Q = k > 1
ht Q = 1 , then either
and let
of prime ideals of
(0)
r .
<J>:D ^
[Y, , . . . ,Y ] -- ► J 1 m
momorphism determined by denote by
U
the kernel
is to show that prime ideal diagram.
andset
H
be the canonical
(Y^) = t^ of
<J> .
for each i
D .
,
The next step in the proof
ht(H^m +n ^ + U)/H^m+n^ = n of
D ^ — ho-
for each proper
Consider the following commutative
and
296 (D/HJIXj,.
J - D[X,, .. ..JL.t.
. •,Xm ,Y1.... Yn ]
(D/HJtXj, . . . . X ^ t *^),...* (tn)]
( D ( m ) ).
Here (D/H)
is the canonical map.
t
(m)
and
0 ^
t
duces an epimorphism
[(D/H) (m:)]
t
of
to
since
t*
of
x(D^m ^) = Hence
n B =
(m) (D
(D/H) (X1 , .
(B) J
(B)
We note that
(D/H) [Xr . . . ,Xm ,T* (tx) , . . .
J
,T* (tn ) ]
.Finally,
(D/H)(m+n)
relations
; we note that part
ker a
has height
H (m)(D (m))
n J = H (m^J .
kernel ^ ( H ^ J ) it follows that fore
T(H^m+n^ + U) has height
(D/H) (m+n)
obtain the desired equality
Consider a chain
of prime ideals of .
primes of first to
Then
of (21.3)
The kernel of
is
has
Considering
as
is the kernel of n.
and
Finally,
a .
an
,
Th ere
x(H^m+n^ + U)
D^m+n^
jjCm+n)^(m+n) ^ fTOm which we
ht(H^m+n^ + U)/H^m + n ^ = n .
We return to the proof that
4>_ 1 (Q)
(1)
(H^m+n^ + U)/H^m+n^ under the canonical iso
morphism between
s + 1 .
is the
determined by the
Therefore
= H^m+n^ + U .
x(H^m+n^ + U)
corresponds to
n .
ij;
onto
( D / H ) h o m o m o r p h i s m of
shows that
onto
,Xm ) , and the restriction
is an epimorphism of
a(Y^) = T *(t^)
,X ) ’ nr
) B = D(Xj,.
in-
t
r
+ht (P^m ^J/P^
p (m+n) < U, 1 of length
J)
^
>
Theorem
But then and this is equivalent to
r + ht (P ^
J/P^,
J) > s +1.
completes the proof of Theorem 21.8. Proof of Theorem 21.4. ment in Theorem D^n ^[{hj}^]
If
We prove the equivalent state
21.6 concerning Q
D^n ^ and
J=
is a proper prime ideal of
J and
if
This
298 P = Q n D , then Theorem 21.8 implies that h t ( P (-n -lJ) + ht(Q/P(n)J) s h t f P ^ J )
ht Q =
+ n < dim
.
This inequality is sufficient to imply the inequality dim J ^ dim
, and the reverse inequality has already
been noted. fk)
J
The prime ideal structure of
has clearly been
used in a strong way in the proof of Theorem 21.8, and hence in the proof of (21.4).
On the other hand, the part of the fkl Rl J
theory pertaining strictly to
in the proof of (21.4)
is confined to (21.1) and to Theorem 21.7. extends part
(1)
Theorem 21.8
of (21.1) to monoid rings
is a finitely generated monoid lying between
D[T]
, where
T
ZQS^@...©Zsn and
ZSi@...@Zsn . We proceed to extend each part of (21.1) to mon oid
rings
sion— free. chain in R[S]
R[S] , where
S
is finitely generated and tor -
To accomplish the extension, we define a special
R[S]
as a chain
Pq g P^c...cP^
with the property that
chain for each
i
between
THEOREM 21.9. that
S
of prime ideals of
(P^ n R) [S] and
0
Assume that
is a member of the
k .
R
is finite— dimensional and
is torsion— free and finitely generated.
a proper prime ideal of
R[S]
ht Q = ht(P[S]) + ht(Q/P[S])
and let .
Let
P = Q n R .
Moreover,
ht Q
Q Then
can be
realized as the length of a special chain of primes of with terminal element Proof.
be
R[S]
Q .
As in the paragraph following the proof of
Theorem 21.5, there exists a finitely generated submonoid of
S
such that
S
is integral over
basis for its quotient group
, and
T , T
contains a
G^ = G - Zn
for
T
299 some
n e Z+ .
Let
= Q n R[T] .
ht(P[S]) = ht (P[T]) , that and that ht(Q) = ht(Q^) multiplicative system
We show that
ht(Q/P[S])- ht CQX/P[T]J
.
Since
P[S]
does not meet the
{Xs | s e S} , we have
ht(P[S]) = ht(P[G])
= h t ( P ^ ) ;by the same argument,
h t (P[T]) = h t ( P fn))
, and hence ht(P[S])
Qf = Qj/P[T]
.
9 it: suffices
ht Q* = ht
In that case we have
R[S]
and
ht Q* = n -
k 2 = tr.d.
[R[T]/Q*]/R .
ht Q* .
k^
overR , and Since
follows that
R
is a field.
ht
R[S]/Q*
ht Q* =
Q-. = ht(P[T]) +ht Q* .
1
W 0 < Wj^
Since
and
(R/P)[X]
the degree of any nonzero element of multiple of
prime ofR
a^
is a
, it follows
a^ {
P
while
is a unit of
R
and
i > 2 .
We show that the conditions given on
is both injective and bijective.
g(f) = 0 .
To show that
f
Thus,
g = 0 , it suffices to
g(X - f Q) = 0 , for the substitution mapping
yY r is clearly an automorphism of R [X ] . Hence, we 0 assume without loss of generality that fg = 0 . Let g(X) = Z ^ g . X 1 . Assume that
The constant term of
g(f) is
= ... = g^ = ^ = 0 , where i £ m
g(f) = f1 Cgi + g i+1f +•••+ gm fm ’1) • however, that g- + g-_Llf -i 6 i+l
f
follows that
g = 0 , and
and g. = 5i y^
R[X]
assume without loss of generality that
k , the order
of
R .
If
We assume that the inclusion element
f
where
r >
where
B^
e R [X ] 2. has
fg = 0
R[X] c R[f]
f = X A[X]
it
R[f] =
y^ , we can
of nilpotence of the ideal
k = 1 , then
0 =
, and hence
Since
R[fj*(f - fq)] , in proving surjectivity of
on
Then
0 .By induction 7
is injective.
We then establish the inclusion
0 .
Corollary 8.6 shows,
is not a zero divisor in
+. . .+ g fm_1 = 0 5m
.
gQ =
and
f^ = 1 .
by induction B^ = (f2 ,...,fn )
and the inclusion is clear. c A [f ]
of the desired form
for
holds for each which k < r ,
Consider h = X + h 2X 2 + '**+ hvXv € R[X] order of nilpotence
r . For i
>2 , note
,
305
that
h^h1
has the form
v ij 6 ^Bh ^2 * has the form
T h u s > if X
=
, where each
g = h “ h 2h2 “ • • '“ hvhV ’ then
+ g 2^ 2 + -• • + gm Xm » where each
It follows that
Br l c B?^r ~ ^ g “ h
hypothesis implies that R [g ] s R[h]
h^X1 +
R[X]
e
g
(B^)2 .
= (0) , so theinduction
c R[g] .
Since the inclusion
is clear, this completes the proof.
Some of the ideas in the proof of Theorem 22.1 are u s e ful in determining case
n = 1
AutRR[Zn ]
as well.
We consider the
separately, not because it is necessary to do
so, but primarily because the notation is much simpler in this case.
Thus, we regard
R— automorphism
of
R [Z ]
R[Z]
as
maps
X
R[X,X *] to a unit
R[X,X ■*■] , and is completely determined by (X) = u , then u e R[X,X"*] map
^(Ea^X1) = Ea^u1 .
is a unit, then the
f(X) — ► f(u)
of
R[X]
R[u]
has a canonical
R[X] ^ i y» = R[X,X
Ea^u*
f(X) e R[X,X *] .
R[X,X"*]
that maps
of (22.1).
X
of
Conversely, if
, and the extension maps
$ the
R[X,X *] , we denote by
u
u ; to wit, if
R[u ]|ui^«> = R[u,u *] for arbitrary
An
substitution
onto
extension to a homomorphism of
.
onto
f(X) =
For a unit
Ea^X1 to
u
R— endomorphism of
to u .
Theorem 22.3 is the analogue
The proof of Theorems 22.3 and 22.4 use the
following auxiliary result. THEOREM 22.2. prime subring of the ring ntu.u'1 ,!!] .
Assume
n . R .
Let
Then
that u be a
u — n
R
is a unitary ring with
unit and
n
be a nilpotent
is a unit of the ring
of
306 Proof.
Assume that
= 0 .
s.. „ w rk-l i k-l-i. (u - n ) C Z i= Qu n ) , so
which is in
IT[u"^,n]
THEOREM 22.3. that
u =
R— endomorphism of
u^ = u ^ — n^ =
f
.-1 „k-l i - k k - i -1 (u - n) = £i = 0u n
,
.
Assume that
=
Then
R
is a unitary ring
is a unit of R[X,X
. Let
R[X,X~*]
X
that maps
<J>u
to
and
be the
u .
The
following conditions are equivalent. (1)
u
is an automorphism.
(2)
u issurjective (that
(3)
ui is
Proof. follows from We write
nilpotent for
(1) (2): (2)
.
generated by
r^ e R^
and hence (2)
where
u^ .
R ^ X j X -1] = R ^ U j U " 1] .
Noetherian ring
let
a
2i=mriul = ®
X =Za^u1 , X~* = Zb^u* , u -1 = Zc^X^ R
$u =>
R^[X,X"*] for each
Then
.
.
We need only prove that
cients r^ , a^ , b^ , c^ , and
Since
R[X,X *]= R[u,u *])
i { {- 1,1}
Assume that
the unitary subring of
and
is,
(1)
9 where
r. e R .
. Let
R^ be
1
and the coeffi
R^
is Noetherian
The restriction of
<J>u
to the
issurjective, hence injective. i , it follows that each
r^ = 0 ,
is also surjective. (3):
Let
P
be a proper prime ideal of
be the canonical map of D = R/P .
Then
R[X,X-1] onto
R
and
D[X,X-1] ,
D[X,X_1] = D[a(u),a(u)~1 ] .
Since
Z
is torsion— free and cancellative, Theorem 11.1 shows that a(u) where
is a trivial unit of d
is a unit of
and this implies that
D .
D[X,X_1] --- say Thus
k = ±1 .
a(u) = dX^ ,
D[X,X"*] = D[X^,X"^] It follows that
u^ e P
, for
307 i
{- 1 ,1} , and hence (3) = >
(2):
ik
is nilpotent.
We assume that
(3)
is satisfied.
u 1 = Z? i= cv.X* i
.
tent and that
R = (u^)
integers
We first establish the inclusion
q .
Theorem 11.3 shows that © (u^)
u
Let
has unit con-
for all sufficiently large
R[X,X *] s R[u,u_1]
under the additional hypothesis that
u^
and
is a unit of
ideal of
R
i * 1 .
Then
R
u ^
is nilpotent.
generated by the cofficients Bu
u = u^X
the case where If
a
and
B = Bu
a(u) = u*X , where
(u|)'1X 1 .
It
each integer
n
.
and
t
unit of r~* . n
R For
over the range
be the
u , where
n
.
of
Bu .
R[X,X"*]
to
k > 2 .
(R/B)[X,X *]
e B[X,X"'I‘]
for each
n .
r Let
n
for
is a s„ = n
u.s.u* = u.X* i i i
v = u -
a £ i £ b , then
> taken over
2 -1 v = u ^X (mod B [X,X ])
and the induction hypothesis implies that To complete the proof for this case potent), we need to show that
,where
v
(u^
R[X,X a unit,
is a unit of
c R[v,v *] . u ^
nil-
R[u,u *] ;
this follows immediately from Theorem 22.2. We observe that the case just considered also handles the case where
u^
is a unit and
u1
is nilpotent, for
the proof shows that under these conditions the cofficient of
X
in
u "1
,
a(u *) =
un = (i^X)11(mod BtX.X-1])
un = r Xn + t n n
Hence if
If
Consider
u* = u^ + B , and hence
follows that Thus
k
has order of nilpotence
a < i < b , i * 1 , w e have
(mod B 2 [X,X *])
of
R[X,X"*] = R[u,u *] .
is the canonical map of
then
u^
Bu
is nilpotent, and to prove the result, we
use induction on the order of nilpotence k = 1 , then
Let
is a unit and all other coefficients are
308 nilpotent. u i s
The remaining case is that where neither
nilpotent.
Let
e^ and e2
sponding to a decomposition u = e^u + e 2u , and where
e^u *
Re^ = (u?)
e^u
in
R .
Re^[e^u,e^u *]
As an element of
e^X
Then
, and
Re1 [e^X,e^X*1 ] ,
e iu i
in e^u is
, and the coefficient of
nilpotent element.
of
R ^ u " * ] = Re^[e^u,e1u *] © R e2 [ e ^ . e ^ 1] ,
e 2u_1 •
the coefficient of
idempotents corre
R = (u^) © ( u ^ )
is the inverse of
similarly for
be
nor
> a unit of
e ^ X i s
e iu _i > a
Since all other coefficients of
e^u i
are
nilpotent as well, it follows from the case considered above that
Re^fe^Xje^X 1] = Re^[e^uje^u-*]
ment,
.
By a similar argu
R e 2 [e2X , e 2X *] = R e 2 [e2u,e2u~* ] .
Therefore
RtX.X'1] = Re1 [e1X,e1X' 1] 9 R e ^ e ^ . e ^ ' 1] = Rfu.u'1] ,
and
this completes the proof of Theorem 22.3. In order to pass from
AutRR[Z]
from Section 11 that a set
to
AutRR[Zn ] , recall
E = {e^,...,e^}
of nonzero
lr
orthogonal idempotents of split the unit
u e R[S]
vertible elements t e R[S]
R[ X±gl ,...,X±gn] ring in of
n
conversely, of
if
1 =
is said to
if there exist
u = vCE^e^X^i) + t .
Zn .
We regard
t where
variables over
R[Zn ] , each
with
£i>* ••>£]< 6 ^ » and a
such that
a free basis for
R
Xgi
R[Zn ]
R[Xg l ,...,Xgn] R .
Under an
maps to a unit
u = (u^,u2 ,...,un )
R[Zn ] , there exists a unique
u^
is an
a unit
v e R , in
nilpotent element Let
be
as is the polynomial R— endomorphism of
R[Zn ] .
And
n— tuple of units
R— endomorphism
<J>u
309
ofR[Zn ]
such that
u (Xgi)
=
the canonical extension to onR[Xg l ,...,Xgn]
u^ for each
R[Zn ]
i ;
u
is
of the substitution map
that maps Xgi
to
u^
for each
i .
As in (22.1) and (22.3), we are therefore left with the problem of determining conditions on the (u^,...,un )
in order that
u
n— tuple
u =
should be an automorphism.
This is where the notion of splitting comes in; Theorem 11.16 shows that there exists a set splits each
of idempotents of
R
that
u^ .
THEOREM that
E
22.4.Assume that
u = (u, ,...,u ) 1 n
preceding paragraph.
and
R is a unitary ring
d> Tu
and
are as described in the
The following conditions are equivalent.
(1)
u
is an automorphism.
(2)
<j>u
is surjective (that is,R[Zn ]
=
RfuJ1 ,...,!!*1 ]) . (3)
Let
E =
idempotents of splits each
R
u^ .
t^ +
he a set of nonzero orthogonal such that For
Re^
.
and such that
1 £ i < n , write
» where
is a unit of
1 =
t^ e R[S]
Then
E
u^ =
is nilpotent and where
{tKj >1 = 1
a basis °f
u^
f°r
1 < j < m . (4)
There exists a set
E
u^
is as represented there, then
Zn
for
as in
(3)
{bjj ^i=i
such that if each a basis of
1 < j £ m .
Proof.
(1)
Thus assume that
(2): We show that
Z r ...... u*1. ..uln = 0 X1
xn
1
(2) implies .
(1)
.
As in theproof of
n
Theorem 22.3, there exists a Noetherian unitary subring
R^
310 of
R ±
containing each
g
±
r.
.
such that
g
R-^fX
n ] = R-^[u*1 ,...jU*1] .
tion of
<J>u
to the Noetherian ring
hence injective.
Therefore the r e s t r i c
R-^[Zn ]
It follows that each
is surjective,
r.
= 0 , so • ,xn
<J>u
is also injective. (2) = >
(3):
1 < j < m , let N
u
•
of
R[Zn ]
Cj If
is a unit of
Let
N
be the ideal of Vjj = u ^
R^ = R/C^
to
be the nilradical of
+ Cj
for each
±b,. ±b . 1J,...,X n:>]
R[Zn ] ; see Theorem 9.9.)
That
Zn , and since
Zn
is a basis for (3)
(4) = >
v^.
b , . ±1 b . ±1 3 ) ,...,tvnjX n j ) ] =
(We have tacitly used the fact that
(bij>i_i
i , then
Rj[Zn ] , we obtain the relation
Rj[X
generates
generated by
, and under the canonical homomorphism
Rj [Z ] = Rj [ O j j X
of
R
R , and for
implies (2):
.
N[Zn ]
is the nilradical
It follows that
{b^
has torsion— free rank
n ,
Zn .
(4)
follows from Theorem 11.16.
Let the notation be as in
(3) .
R[ X±gl,. . . ,X±gn] = E? =1 © R e ^ e i X * 8 !, . . . ,eAX ±gn]
Since
and since
Rtu*1 ,. . . ,u*X ] = E?ssl © Re._te.uJ1 , . . . ^ u * 1] , it suffices to consider the case where loss of generality that is nilpotent and ideal of
R
v^
m = 1 .
Thus, we assume without
ik = t^ + v^X^i
is a unit of
R .
t where
t^ e R[Zn ]
Let
be the
generated by the coefficients of
Bu
t^,t2 ».-->t
•
311 Then
Bu
is a nilpotent ideal of
inclusion
R , and to establish the
R[ X±gl ,... ,X±gn] c R [u^*,...,u**]
duction on the order of nilpotence then
k
of
, we use in
Bu .
If
k = 1 ,
Rfu*1 ,....u*1] = R[ X±bl,...,X±bn] = R[X±gl ,...,X*gn]
Consider the case where k > 2 .
If
a
a(u^) = v^X^i
aCu^1) = (vt) *X b i .
integers
a^,...,a
a, x
has order of nilpotence
is the canonical map from
(R/B)[Zn ] , then hence
B = Bu
.
, where
R[Zn ] v? =
to + B , and
It follows that for arbitrary
that
a a. a l n. _ •••un E v i •••vn
a,b,+...+a b l l n n / •% d r7^i ^ (mod B [Z J) .
Therefore
O)
X
where
a-,b, +. . .+a b n n =c
c
a, a u1 ...un + f , a 1a9 ...a 1 n a1a 7 ...a 1 it n 1 z n
is a unit of
1 2 **
Next we modify
u^,...,un For
monomials
> where
y^.
Thus,
(*)
w ij € Zb^©...@Zbn .
be expressed in the form
n
to obtain elements
h^,h2 »...,hn , as follows.
R [ u ^ , . . . ,u*n ]
e B[Zn J .
R and f
1 2 *''an
1 < i £ n , t^ e B
is a sum of
and where
implies that
+ P^
, where
is nilpotent, and where
can z^
e
p^. e B2 [Zn ] .
Let
h^ = u^ - 2zij > t^ e sum being taken over all monomials y ^ X w ij hi
in
t^ .
By choice of the elements
c R[u^ ,...,u" ]
where
t* € B 2 [Zn ] .
and
h^
has the form
z^
,
v^X 1 + t* ,
The induction hypothesis implies that
R[Zn ] 9 R[h-^1 , . . . ,h 1 ] , and Theorem 22.2 shows that +1 ±\ R[h^ ,...,hn ] c R[u"
+i
, ...u~ ]
.
matical induction, the equality
By the principle ofmathe R[Zn ] = R [ u ”1 ,...,u ~
follows. The automorphism
group of R[G]
, where
sion— free but not finitely generated, but in the case where
R
Theorem 11.6 shows that
G is
tor
is not known in general
is reduced and indecomposable, R[G]
has only trivial units.
For
any group ring with this property, Theorem 22.5 indicates how the
R— automorphisms of
R[G]
arise.
THEOREM 22.5.
R
a unitary ring with group
units
U(R)
.
Let
Assume
be
that the group ring R[G]
of
admits only
trivial units. (1) UgX^g
If
for
(j) e AutRR[G] some unit
<J>:G — *■ U(R)
, then for each
u^ e R
and some h^ e G .
defined bya(g) = Ug
larly, the mapping
ip:G — ► G
g e G ,
is
a
=
The mapping
homomorphism;
defined by
iKg) = hg
simi
is
an automorphism. (2)
Conversely,
ip e Aut(G) such that
<J>(Xg ) = a ( g ) X ^ g^ (1):
g,,g, e G , then 1 L
, then
is a homomorphism and
for each g
By definition,
augmentation map on
0
a:G — ► U(R)
, then there exists a unique element
Proof.
, where
Hence
g l+g2 if>(X ) = u
y
is the
is a homomorphism. h g l+hg2
+ ^^g 2^ • Xg eR
If
is an
e
at most one
To verify that
2 , the relation i * j .
Then
r ^
- r^. eJ
n
hypothesis,
e J , so
Clearly
RS/I
s^
with
[s 1 = 0
has fewer than f^
B .
. The inclusion
To prove the reverse inclusion, take
^i=lr is i e I\(0) case
RS
x ~ y}
and
n = |Supp(f)|
and
is commutative.
.
The
n = 2 , then = [s2 ]
and
~ s 2 and implies
ri = f eJ
s^ ~ Sj
f 1 = (f - (r ^
elements in its support. f e J
f =
~ r2 * .
For
for some
- r ^ )) € I
By the induction
I =J . Let C be
any ideal of
319 RS
such that RS/C
on
S
by
is
commutative.
x y y if and
only if
It is routine to show that S . We verify that
y
y
Define a relation
rx - ry e C
for all
is a congruence on
x y y , z e S , and r e R
rxz - ryz
and
k Z. ,a.b. J=1 3 J For each
rzx - rzy are
where
a.,b. J’ J we have
j
e R
r e R .
is an equivalence relation on S
by showing
that it is compatible with the semigroup operation. assume
y
Thus,
. We must showthat
in
C .
: this is
Write
r =
possible since H
2 R = R
a.b.xz - a . b . y z = (a.x — a.y)b.z e C , and 3
3
3
3
3
3
3
a.b.zx - a . b . z y = a.zfb.x - b . y ) 3
3
3
3
7
3
3
3
e C
.
v rxz - ryz = E^fa^b^xz - a^.b^yz) e C , and similarly,
Hence
rzx - rzy € C . s,t e S
and if
We show that S/y is commutative. If v r = Z^a^b^ e R , then a^b^st - a^.bjts =
a^.s-bjt -bjt*a^.s e C this implies that commutative. x ~ y .
since
rst - rts
Therefore
Then
x y y
so
RS/C e C , so
y > ~ . Let rx - ry
e
B = {rx - ry | r e R , x ~ y}
Since
lows that
is commutative.
C
As above
st y ts
and
x,y e S
be such that
for each
S/y
is
r e R .
generates
I , it fol
I c C , and this complets the proof of Proposition
23.1. COROLLARY 23.2. R2
are idempotent.
where
~
and
respectively, Then
y
Assume that Let
S' = S/~
R^S = R 2T , where and
let
S
and
T' = T/y ,
are minimal congruences on
S
and
T ,
such that the factor semigroups are commutative.
R ^ ’ = R 2T ’ . If
R^
and
T
are commutative and if
<J>
is an
f 320 isomorphism of isomorphism
R^ [S J
*
where
NCR^S])
R^ [S ]
and
of
onto
R 2 [TJ
* then
R 1 [S]/N(R1 [S])
and
N(R2 [T])
onto
R 2 [T ] , respectively.
N(R1 [S])
istic
.
If
R1
and
R2
are re
p , Theorem 9.4 shows ~p
of
S , and a similar statement holds for
On the other hand, 0
R 2 [T]/N(R2 [T]) ,
is the kernel ideal of the congruence
p— equivalence on R 2 [T]
induces an
denote the nilradicals of
duced rings of prime characteristic that
<J>
if
, then Corollary 9.12
[S]/N(R^[S]) = R 1 [S/~]
R^
of character
implies that
, where
asumptotic equivalence on
is a domain
S .
~
is the congruence of
The next result follows imme
diately from these observations. THEOREM 23.3. semigroups and (1) teristic
If
Assume that
R^S] R^
S
and
T
are commutative
= R 2^T ^ *
and
R 2 are reduced rings of
prime charac
R 1 [S/~ ] - R 9 [T/~ ] , and each of these
p , then
i
p
p
l
rings is reduced. (2) then
If
R^
and
R 2 are domains of characteristic
R ^ [S/~] = R 2 [T/~]
0 ,
, and each of these rings is reduced.
Our next reduction of the isomorphism problem is to the case of periodic semigroups. periodic semigroups.
We denote by
We denote by
S*
elements of the commutative semigroup
S*
the set of
the set of periodic S .
The hypotheses of
results through Theorem 23.16 either assume or imply that S* *
R
an abelian torsion— free group. the ideals
(f)
and
if
S
is a monoid.
is a reduced ring and If
f e R[G]
is such
(f - f ) are idempotent, then
G
is
that f
e R .
321 Proof. u,v e R[G]
Write Let
of
R .
and
f - f2 = v(f - f 2)2
with
g * 0
f = Za Xg . We show that a = 0 for g g by showing that a^ belongs to each proper prime
ideal
P Let
.
f = uf2
be the canonical homomorphism from
integral domain
CR/P)[G] .
R[G]
to the
Then
<J>(f) = (f)2
anc*
♦(f - f2) = ♦(v)*(£ - £2)2 . If
Cf ) = 0 , then
a^ e P .
If
<J>(f) * 0 > then the first
of the equations above implies that
(R/P)[G]
and
If
(j>(f)
then
is a unit of
^(f)
(R/P)[G]
.
(f) = <J>(f2) = <J>(f)2 >
is the identity element of
Supp(<J>(f)) = {0}
and
a^ e P .
is unitary
(R/P)[G]
Finally,
if
; hence
<J>(f)
and
2
<J)(f ~ f ) of
are nonzero, then each is a unit.
(R/P)[G]
But the units
are trivial by Theorem 11.1, so that if
Supp(<J>(f)) = {s} , then
Supp(<J>(f) -(f)2) = (s,2s)
must be a singleton set.
Thus
THEOREM 23.5. that
R[G]
ideals where
(f) G*
and
s = 0 , a^ e P , and
Assume that
is reduced.
If
(f - f2)
Write
Supp(f) u Supp(u) u Supp(v)
mand of H , say
H = H* © W
f , u , v e R[H] = R H * [W]
with
. Since
f e R [G *]
,
as in the
be the subgroup of
H*
such
G .
f = uf2 , f - f 2 = v(f — f2)2 H
e R .
is such that the
are idempotent, then
generated, the torsion subgroup
f
is an abelian group
f e R[G]
proof of Theorem 23.4, and let rated by
G
is the torsion subgroup of
Proof.
, which
. Since of
W R[H*]
H
H
G
gene
is finitely
is a direct sum
torsion— free. is reduced,
We have
322 f e R[H*]
by
Theorem 23.4.
The result then follows since
R[H*] c R[G*] . THEOREM 23.6.
Assume that
lative semigroup such that f e R[S]\(0)
S
R[S]
is reduced.
is such that the ideals
are idempotent,
then the set
S*
is nonempty,
is a monoid,
and
S
Proof.
The nonzero ideal
potent element of
Supp(e)
is periodic. w .
R[G]
S
and
is a quotient ring of
(f)
the torsion subgroup of To
S S
G .
S .
(n + l)s = s , which says
since
S
n > m
w
is an
and is therefore reduced.
Thus
thus
with
contains
The group ring
f € R[G*]
, where
f e R[S]
ns = 0
that
for some
s e S*
.
such that
is cancellative.
G*
is
S n G*
=
n R [G *] =
complete the proof, we check
s e S n G* , then
n,m e Z +
S*
is a monoid.
R[S]
If
and
S
is generated by an idem-
is cancellative,
S* .
(n - m)t = 0
of
.
S* * and
From Theorem 23.5 we conclude that
exists
(f - f2)
f € R[S*]
be the quotient group of
R[S n G*] .
and
of periodic elements
Hence
Because
identity element for G
(f)
If
e , and Theorem 10.6 shows that each element
an idempotent
Let
is a commutative cancel
If
n e Z+
,
and
t e S*
, there
nt = mt , implying Thus
t e S n G*
S n G* = S* . Theorem 23.10 is the last of the series of four propo
sitions beginning with Theorem 23.4 that have basically the same form.
The hypothesis in (23.10) differs from that in
(23.6) in that we no longer assume that
S
is cancellative.
The proof of Theorem 23.10 uses the Archimedean decomposition S = uS„ a
of
S
introduced in Section 17.
We recall from
323 Theorem 17.10 that each of
S
if
S
S&
is a cancellative subsemigroup
is free of asymptotic torsion.
Two preliminary
results are needed in the proof of (23.10); the first of these concerns the Archimedean decomposition of THEOREM 23.7.
Assume that
S
is a commutative semigroup
that is free of asymptotic torsion. Us
= {t 6 S | s + S c (1)
Us
s
= U, t (2)
b +
is a subsemigroup of Sg
for each
t e S
If a,b
e Ssand
c , then a =
S
s
t 1 ,t2 e
c
e Us
in
aresuch that a + c =
(1) that v
U
6Z+ anc*
and
t l + t 2 + S1
+ s 2 6 tl + t2 + S *
k 2s = t 2 + s2 .
s + S c radft^
+ t 2 + S)
the
U s
+S
s -
sl ,s2 6 Then
U
s
u e U
follows since
Write
k € Z+ and
+v + w
To establish
,v e s ’
. s
We
; the inclusion
rad(u + v +S) c radfv + S) kv = u + w ,where
such that
(k^ + k 2)s =
and t^ + t 2 e U s .
c S , take s ’
S
Therefore
rad(u + v + S) = rad(v + S)
(k + l)v = u
S„ c u and s _ s
is closed under addition, take ’
s
kis = t^ + s^
show that
Moreover,
s
Choose
inclusion
containing the
follow immediately from the definition of
that .
S
b.
t e Sg
. To see
s e S , define
of s as an ideal.
Proof.The assertions ----for
For
rad(t + S)} .
Archimedean component U
S .
u + v + S c v + S .
w e S .
Then
implies v e rad(u + v + S) , and
rad(v + S) crad(u + v +
S)
. This completes the
hence
proof
of
(1) . (2) (1)
: Since
shows that
a + c = b + c , then c + a e S
s ’
, and S
s
a +(c + a) = b +(c is a cancellative
+ a)
.
324 semigroup since fore
S
is free of asymptotic torsion.
There
a = b . THEOREM 23.8.
group and that
p e Z+
congruence on (1)
Assume that
S
is prime.
is a commutative semi Let
~
be the cancellative
S .
If
S
is free of asymptotic torsion, then so
IfS
is
is
S/~ . (2)
Proof.
p— torsion— free, then so is
We recall that
if and only if
[b]
(2)
nb
= [mb]
+ x
m(b
.
and
ma
x e S .
n , m such that
+ y = mb + + x + y)
+ y = b + x
torsion.
9.17 shows that reduced,
is not
since
Consequently,
Assume that
(iii)
+y
(ii)
p
S
where
~
(iii)
and
na
+ x
= =
such that
y . Therefore
m(a +
x+ y)
= n(b + x + y)
, and
a + x + y
Hence
Sis free of asymptotic ~b
and
[a] = [b] . Theorem (i)
R
is
is free of asymptotic torsion, and
p— torsion— free. and
[nb]
is reduced if and only if
is regular on
(ii)
We prove Assume that
is a commutative semigroup.
R[S] S
a
a ~ b
Choose rela
[na] =
b + x + y are asymptotically equivalent.
a +x
by
is similar, but easier.
There exists x,y € S
+ x + y) , n(a
and
for some
.
S
e S/~ are asymptotically equivalent.
tively prime integers [ma]
is defined on
a + x = b + x
(1) ; the proof of [a],
~
S/~
R
for each prime
p
such that
S
Theorem 23.8 shows that properties
transfer from
R
and
S
is the cancellative congruence on
obtain the following corollary.
to
R S .
and
S/~ ,
Thus, we
325
COROLLARY 23.9.
Let
~
on the commutative semigroup so is
be the cancellative congruence S .
If
R [S ]
is reduced, then
R [S/— ] . Corollary 23.9 is the final preliminary result needed in
the proof of Theorem 23.10, which will prove to be a key r e sult in the ultimate proof of Theorem 23.17. THEOREM
23.10.
group such that
R[S]
that the ideals
(f)
S* *
and
is a commutative semi
is reduced. If f e R[S]\(0) is such 2 and (f - f ) are idempotent, then
The proof that
in the proof S
.
is the
same as that given
Let S = uSa
be the decompo
into its Archimedean components, and let S
Since
if and only if
R[S]
is reduced,
totic torsion, and hence each
S&
rad(a + S) = S
is free of asymp
is cancellative.
the result, we use induction on the cardinality equivalence classes under Supp(f) If
n
m
To prove of the
represented by elements of
. m = 1 , then
f e R [Sa l .
Supp(f) c S&
a e S
for some
and
To verify the theorem in this case, it is suf
ficient, by Theorem 23.6, to* show that generate idempotent ideals in
f
and
Toward this end, we write
f - f2 = v(f - f 2)3
for some
f , f3 , f - f 2 , and
(f - f 2)3
u = u.+...+u , where x r
f - f2
R [Sa l > f°r then
f e R [S *] c R [S * ] .
Write
n
that gives rise to this decomposi
tion --- that is, a n b + S)
S* * <J>
of Theorem 23.6.
be the congruence on
rad(b
S
f e R [S* ] .
Proof.
sition of
Assume that
u,v e R [S ] .
f = uf3
The supports of
are all contained in
Supp(u-) c S l a
and
Sa .
and where
326 *S
S ai
for
i* j .
Then
r 3 = E,u.f 11
f
aj
Supp(u.f^) s S . i a^+a
It follows that
sum being taken over those over the same indices 2
f =(Zu. f)f ideal of
.
a
such that
Therefore
f
, the
j
Sa +a = Sa . Summing aj Eu.f e R[S ] and j a
generates an
idempotent
2
R [Sa ] •
By the same argument,
rates an idempotent ideal of the case
f =
j , we note that
2
e R[S]f
J
j
, where
f - f
also gene
R[Sa ] , and this establishes
m = 1 .
We assume that the result is true for
m < n , where
n > 1 , and consider the case where the elements of determine tains b +
n
classes under
an aperiodic element S £ r a d ( s + S)
there exists taining
s
R[P]
b
R[S]
is an ideal and
.
s e Supp(f)
Consider
; here of
f - f
idempotent ideals of
the
S\P
.
If not, then
2
.
Let
Then
, so
R[S\P]
P
of
n
implies that
f^ e R [S *] ,
b e Supp(f^)\S* asserted. ment of
since
f^
.
is the
and
^
- f^
R[S]
.
The number of
It follows that Supp(f)
that b
S^contains
and thatSupp(f)
{t e S | b + S c rad(t + S ) } .
generate
Supp(f^)
The induction hypothesis
and this is
We conclude
S
f = f^ + f 2 ,where
fj -
and of
Sg c P .
con
decomposition R [S ] =
Archimedean components represented by elements of is less than
S
is a subsemigroup of
R [S ] .
f 2 e R[P]
R[S\P]— component of
Supp(f) con
b . We first show that
for each
but not
f 1 € R[S\P]
Assume that
s e Supp(f) and a prime ideal
R[S\P] + R[P] of and
n .
Supp(f)
c
Define
a contradiction,
for
+ S c rad(s + S)
, as
each aperiodic
ele
=
327 We show that the assumption that tradiction. where
Write
onto
S
S/~ .
Moreover,
and let Theorem
<J)(f)
ar*d
idempotent ideals of I = , then
[b] e
23.6, or else s € Supp(f^)
.
Let
f2 e
~
anc*
be the cancellative
be the natural homomorphism of
23.8 shows thatR[S/~]
is reduced.
2
(f - f ) = 4>(f) ~ <J>Cf) R[S/~]
.
generate
Consider [b] e S/~ .
Since
for c e Supp(f2) , c * b .
Supp(<J>(f))
[b]
and [b] e (S/~)*
{ Supp(<J>(f))
and
There
by Theorem
[b] = [s]
for some
[b] = [s] e (S/~)*
In the latter case,
s e S* .
since
4>
[b] * [c]
fore either
is empty leads to a con
f = f^ + f2 , where
Supp(f^) , and
is reduced and
let
f e R[T]
generate idempotent ideals of
that the definitions of
I
and
is such R[T]
T
.
imply
J =
.
is a
.
The ring
such that
S
.
are commutative semigroups with
reduced, and
f
There
is the unique maximal
contains each regular subring of
COROLLARY 23.13. T
R [S ] .
S* * <J> , and R[S] is reduced. 2 f - f generate idempotent ideals
Theorem 23.10 shows that
W 2 R [S *]
and
generates
is a regular ring,
the unique maximal regular subring of
S
f
f e I [S ] .
is regular.
THEOREM 23.12.
Let
it follows that
e .
implies isomorphism of
periodic commutative semigroups
S
and
F T
and .
K
for
To handle the
330 case where
S
is not a monoid, we need one preliminary-
ring— theoretic result. THEOREM 23.14. let
P
Let A be an ideal
be a proper prime ideal of
(1)
P is an ideal
for some prime ideal (2)
If
of
Q
R
such that Proof.
rx c A
implies
r e R
rxP c P .
P
is prime in
hence
P
is an ideal of R .
containing
P
i .Hence and
x e A
, it follows that
Q (2):
primes of Then
R
Q
with
and
m < n
A/Pm - (A + Q )/Q m Mn nil
Pm + i = Qm + i n ^ taining
maximal ideal of R
ideals of (2)
{M^ n M} M If
*
(2)
Assume that
R .
distinct from (1)
A/Pm .
f°r some prime
THEOREM 23.15.
of
A\P
rP c A .
Q
in
R .
Let
Q, dim A .
There exists an ideal
such that
is prime in
A ,
rP c P , and
failure to meet the multiplicative system ideal
dim R
x e A\P
Moreover,
Since
R
A
n
,so
is
.Moreover, since F
by elements
S^ is usatis
= 0 , so either
Therefore
a
F is natu
un - um = 0 , where
um (un-m - 1)
is a root of unity.
F[S^] .
n,m e Z + u
= 0 or
is generated over
F
by roots of unity. Let 0
<J> be an isomorphism of
induces a bisection
of
F[S] Aand a
some
a
F
K[T]
K
K .
K
K
of roots of unity.
lows
u^
is
that
n^
.
a(F)
^ a ^aeA
in such a way
Since F - A
is imbeddable in
Moreover,
accomplished by isomorphisms such a way that
Then
a
for
is imbeddable in each residue field
, it follows that
is imbeddable in
of
K[T]
*
F[S]
F . Similarly,
these imbeddings may be
a:F --- ► K
is generated over
and cr(F)
ip:K --- ► F by a set
in {u^}
Assume that the (multiplicative) order Since
iKu^) e F
contains the rrth
has
order n^ , it fol
roots of
unity.
333 Consequently, fore
u^ e a(F)
for each
i
and
a(F) = K .
There
F - K . Unlike the results in this section since Theorem 23.3,
the semigroups in the statement of Theorem 23.17 are not assumed to be commutative. THEOREM 23.17. T
are semigroups,
FS a KT .
Then
Proof. isomorphism where S^
S^
Assume S
F and
fields,
S and
contains a periodic element,
and
F = K .
Corollary 23.2 and Theorem 23.3 show that the FS = KT
and
T^
induces an isomorphism are commutative,
is a homomorphic image of
23.13
K are
then shows that
T^ * <j>
F[S^]
F[S^]
is reduced, and
S .
Hence
and
F [S J ] - K[TJ]
Theorem 23.16, we conclude that
= K [T ^ ] ,
Sj ^
Corollary .
Applying
F - K .
Beyond Theorem 23.3, the material in this section appears to have slight application to the isomorphism question in the case of a commutative semigroup
S
such that
S* = .
The
next result resolves what is probably the most natural sub case of this problem. THEOREM 23.18. and Then
T
F and
are subsemigroups of
Z+
K are fields and
such that
S
F[S] - K[T]
.
F - K . Proof.
and
Assume
Let
m
and
T , respectively.
n
Define
U = {f e F [S] | for each more than
be the smallest elements of
k
k e Z + , f^ elements of
is not a product of F[S]}
.
S
334 We show first that order
m .
f e U .
If
U
is the set of elements of
fe F[S]
has order
that
rd e S
for each
d
k eZ +
Forthis
, and
the set of elements of the set V =
V
more than
above shows
that
<j>(aXm )
the form
<j)(a)
has
of order
n .
we show that the mapping
a^,a2 e F
n
rd
{ U. Similarly,
k elements ofK[T]}
We define
is the coefficient of
u ^be an isomorphism of
q e K[T]
is of the form
, where
(fe K[T] | for each
Let
F[S]
K [T ]
e F[S] has order
k , we have g^ = (Xm )^+1h ,
some
h e
such
such that
h
Hence
choose M e Z +
g
where the degree of each term in r > M .
be the greatest c om
and
r > M . If
w > m , then there exists kw - (k + l)m > Md .
S
of
m , it is clear that
To prove the converse, let
mon divisor of the elements of
F[S]
Xn
in
t:F -- ► K
an isomorphism of
F
.
Then the * 0 .
K\{0}
--- ► K (aXm ) •
If
defined by
onto
K
Then
and some as follows:
(Xm ) = u X n + q
Then
(jiffa-^ + a 2)Xm ) = (bj^+
oCaj) + a(a2)
and
+
+
^2
’ so ° ^al + a 2^ =
+ a 2) = t U j ) + x(a2) .
Moreover,
335 *Ca1a 2X 3m) = i(i(a1a 2Xln) (uX) =
u 1 rX - (HpX1 r ) , where (pX1 r) e T[Z]
Let
e^X
that
*
is nilpotent.
More
deg <Jk = 0 .
—1 T (X) = uX + p , where
Write
a (X) = uX .
is a
^(e^) = f^
Thus, we assume without loss of generality that mentary.
2 ,
.
© Re^ i
v = ue
Theorem 24.2 shows that there exist decomposi
R =
for each
Letting
of
R
and
are subisomorphic.
R [Z ] T , it
A proof of
this statement uses the following result. THEOREM 24.4. the ring a unit of
where
R , let
Let r
X
and
Y
be indeterminates over
be a positive integer, and let
u
be
R .
(1)
R[X,X~ ^ ,Y]/ (Yr - X) = R[Z]
(2)
RtX.X'-’-.Yj/fY1' - u) = R[y] [Z ]
{l,y,...,y
r *1
. , where
yr = u
and
} is a free module basis for R[y] over R.
341 Proof. (1): x = X + I . {y1 }”^
Let
I = (Yr - X) , y = Y + I , and let
We show that
is free over
R[X,X 1 ,Y ]/I = R[y,y *] , where
R .
R[X,X *,Y]/I = R[x,x \y ]
We have
y r = x , a unit of
. Moreover,
x A = (y *)r , so
—X
i ® x,x € R[y,y ] . To show that {y }_oo is free over v it suffices to show that r^ + r^Y+.-.+r^Y belongs to only if each
r^ = 0 .
R , I
Thus, write
E q I^Y1 = (Yr - X)f(X,Y)/Xm , where
f(X,Y) e R[X,Y]
and
m > 0 .
Substituting
X = Yr
k yields that
^oriY
R[y,y 1 ] = R[Z] (2):
let
= 0 * an(* ^ence each
As in
x = X+ J
and to prove
.
(1) , let
that {xxy^
R .
Thus, assume
It is clearthat
|i e Z q
, y =
that
Y + J , and
R[X,X_1,Y]/J “ Rfy.x.x'1] ,
R .
and
{x1y-^ | i e Z
0 < j < r - l }
r - 1
is free over are polyno
such that
+ £X (Y)X+...+fs (Y)XS eCYr - u) , say
Z®fi (Y)X1 = CYr - u)h(X,Y)/Xk , so that E®£i (Y)Xk+i = (Yr - u)h(X,Y) Considering divides each
as
R[Y][X]
, it follows that
f^(Y)
in
R[Y]
Since
each nonzero multiple of
Yr - u
By choice of the polynomials f^(Y) = o
.
R[X,Y]
for each
i .
.
and
In fact, it is enough to
fQ (Y),...,fg (Y) e R [Y ]
of degree at most f0 (Y)
J = (Yr - u)
is free over
show
This implies
, as asserted.
(2) , it suffices to prove that
0 < j < r - 1}
mials
r^ = 0 .
Yr - u
Yr - u
is monic,
has degree at least
f^(Y)
r .
, it then follows that
This completes the proof.
34 2 THEOREM 24.5.
Assume that
mentary isomorphism of degree ring
<J>:R[Z] -- T[Z] r .
is an ele
There exists an extension
T[v]
of T such that vr is a unit of r -1 T,{l,v,...,v } is a free basis for T[v] over
T , and
R - T [v] . Proof.
Without loss of generality we assume that
<J>(X) = uXr , where
u
is a unit of
<J>*:R[Z][Y] -- ► T[Z][Y] ring induced by
T .
Let
be the isomorphism of the polynomial
--- that is, *(Y) = Y
Let
I
be the ideal of
let
J = *(1) = (Yr - uXr ) = C(X'1Y)r -u)
class rings
R[Z][Y]/I
an isomorphism
a
and
for
over
T .
R * T[v]
COROLLARY 24.6. then
R
subring
and R^
such that T
where
T of
The residue
Moreover, the proof of
v
also shows that T[Z][Y]/J v= X ^ Y + J , x = X + J , r —1
}
= vr =
is a free basis
a (y) = *(Y) + J = Y + J =
is a unit of
T[v]
1 , and part
(1)
.
Consequently,
of Theorem 24.3
. If
R[Z]
and
T[Z]
are subisomorphic. T
and
are isomorphic under
{l,v,...,v
We have
is elementary of degree
shows that
.
- X,
R[Z][Y]/I = R[y,y *] s R [Z ] , where
T , and
XX *Y + J = vx , where a
<j>* .
The proof of (24.4)
is a unit of T[v]
T[Z][Y]/J
induced by
T [v] [x,x_1] * T [v] [Z ] , u + J
IR [Z ] = ^ *
R[Z][Y] generated by Y r
Theorem 24.4 shows that y = Y + I .
and
are isomorphic,
In fact, there exists a
and an injective endomorphism
a(T) c R^ c T , where
R^ = R
and where
are finitely generated free modules over
a(T)
a
of R^
and
T and R^ ,
respectively. Proof.
It suffices to show that
T
can be imbedded in
343 R
by means of an isomorphism
generated free module over an isomorphism, T=*T1@...©Tn i
sion ring
of
R and
T^[v^]
<Jk
T
(T) .
Let
T^[v^]
R
<J>:R[Z] -- >-* T [Z ]
be
R = R ^@ ..,0Rn
and
, the restriction of
over T^ .
such that {1,v ^ , ...,vTi
Let
[Z ] for each
(R^ [Z]) =
Theorem 24.5
is a unit of
such that
and choose decompositions
and such that
elementary.
t
x
R^ = T ifv iJ » v i*
= ui
is a free basis for
W = T^[v^]©...@Tn [vn ] , let v =
(v^,...,v )
, and let
bedding
of T
in
{l,v,...,vr
*} is
a free basis for W
a:W -- >-* R
is the natural isomorphism, then t = ay:T ----►
y
r = sup{r^}^ . W , we have
Under the natural im
W = y(T)[v] over
, where
y(T)
.
If R
has the desired properties. We next present an example of non— isomorphic two-dimen sional Noetherian domains T[Z]
are isomorphic; EXAMPLE 24.7.
the Let
R
and
T
such that
a = (1 + / = T 7 ) / 2
, let K = Q(a)
D = Z[a]
Then
is a Dedekind domain with class number
be the integral closure of
is the nontrivial automorphism of
K , then
unique nontrivial automorphism of
D
<j>
Z
be the maximal ideal
show that that of fore
I
.
is not principal,
I5 = bD , where D , we have I2
(2,a)
is not equivalent to
(I *) = I . > where
There
i = 1,2
and
344 j = ±1 .
Let
R =
I2nXn
subrings of the group ring sequences D .
{I2n}°° n=-°°
Since
1,1
D-modules, over
-1
,I
-
, and
I R
Noetherian.
sional Noetherian domain dimension at most
2 .
2
be the Rees corresponding to the
{In }°° n=-°°
it follows that
D , hence
T =
D[X,X *]
and 2
and
of fractional ideals of are finitely generated
and
T
are affine domains
As overrings of the two-dimen
D[X] , the domains On the other hand,
two-dimensional quotient ring of both
R
R
and
T have
D[X,X *] and
is
T , so
a
dim R =
dim T = 2 . Assume that
R
and T
are isomorphic
Under localization at
Z \ {0> , a
K[X,X
a(K) = K , a|K = <J>*
and
.
a(X) = cXJ , where
j = 1 a
Therefore
or
- 1 .
c a
for
i = 1
I2X
onto
or 2 ,
K
and
I^X-* , since
is surjective and since it maps monomials of degree
a(I2)-cXj = ^ C l ^ c x j implies that
.
and
(Y)
= bX 5Y 2 .
are isomorphic.
t
1
t(R[Y,Y 1 ]) c T [ Y, Y_1]
T [ Y , Y -1] . {I2nXnY^ {In XnY-®
Since
| n,j e l ) | n,j e Z}
Hence
t
Let
R
and
t t
be the
(X) = X 2Y , 1 (X) =
We observe that
and that
t'1 (T[Y,Y'1]) c RtY.Y"1] ;
is an isomorphism of
R[Y,Y_1]
t
determined by
is determined by
b " 1x'2Y , t " 1 (Y) = b 2X 5Y ' 2 .
this will imply that
a(I X) =
, which
, a contradiction.
KfX*1 ^ * 1]
Then
On the other hand, (f>i (I2)=
n
2
We show that the group rings
T[Y,Y *]
K— automorphism of
.
Therefore
I2 e
(1) , it suffices
R[Zn ] = T[Zn ]
Theorem 24.7, that
, for
(2)
that
part
.
H
Z [G ] -
is
For is a
g ----- ► h(g) zero mapping, then Z [G ] - R .
If
(Z[G]) £ R
h * 0 , then
a direct summand of Z[K][Z]
is
G -----
Z— invariant by
say
is
347 Theorem 24.8, and hence THEOREM 24.11. radical
J(R)
Z [G ] ^ R .
Let
R
be a unitary ring with Jacobson
and nilradical
N(R) .
Let
S
and S*
be as
in the paragraph preceding the statement of Theorem 24.8, (1)
If
R
is a
(2)
If
R
is an integral domain and if
then
R e S* (3)
R c S*
then
R e S* . J(R)* (0) ,
.
If
R is indecomposable and
minimal prime (4)
field,
P
of
R
such
if there exists
that R/P e S* ,
then
If
R is indecomposable and
if
If
R is quasi— local, then
R e S* .
a R e S*
J(R) * N(R)
.
, then
.
(5) (6)
If
dim R =
(7)
Each of the
0 , then
R e S* .
classes
S*
and S
isclosed under
taking finite direct sums. Proof.
We assume throughout the proof that
:R[Z] -- m - T[Z]
is an isomorphism.
composable in each of cases implies that the ring
T
The ring
R
is inde
(1) — (5) , and Corollary 10.17
is also indecomposable.
Thus
<J>
is elementary in each of these cases --- say
(X) =
uXr + t , and
= 1 . We begin
ourobject is to show that
|r|
to consider the individual cases. (1):
If
R
is a field,
tegral domains, and hence T[Z]
a - 1
R = GF(2)
(a) = vXm
.
If
We show that IRI > 2 , take
for some unit
is a unit of
and
T[Z]
is also a domain.
has only trivial units.
is clear if Then
T
then R[Z]
R , either
v e T
.
vXm + 1
Therefore
(a)e T
.
In either case we conclude
Thus
T[Z] =
<J>CR [Z ]) c T[uXr ,u lX r ] = T[Xr ,X r ] ,
and this implies
that
Ir | = 1 . (2):
As in (1) , (R) c T
in
e T
with
R .
(r) e T\{0}
.
T[Z]
<j>(r) e T
.
The proof that
J (R)
Since <J>(y)
|r| = 1
,
and
Then yr e
,
c T[Z]
is an integral domain, it then follows that
(P[Z])
Consider the minimal prime is minimal in
some minimal prime isomorphism since
(4): P
and
is also a unit of
y e
is a
=
Consequently,
(j)(J(R)) £ T . Take any element
1 + r u e T
2 <J>(l + r + r )
uXm + (uXm - l)2 = uX 2m - uXm + 1
so
Then
for some unit
Moreover,
and this implies that
.
We first
of
R
*
is
of
of
, and hence
T
.
Since J (R)
onto
But
* N(R)
of
R[Z]
<J>
Then
since
for
induces an |
*|= 1
.
<J>(P[Z]) = Q [Z ]
The isomorphism
(R/P)[Z]
elementary.
R/P
e S*
, there exists a minimal
.
prime
that is not expressible as an intersection of m a x
imal ideals of Jacobson
Q
T[Z]
P[Z]
R .
Hence
R/P
R/P e S *
radical, so
is a domain with nonzero by
(2) .
R e S*
Whence
by
(3) . (5):
Let
M
be the maximal ideal of
the result follows from minimal prime of (6):
R
(4) .
R .
In the contrary case,
and the result follows from
In this case
| = |* |
R/M e S* , we (7):
have
We
for some since
Let
is in
for each
phism
e^
i
be
(Z?u.)X + 1 l zjt. e T[Z]
onto
S*
R.
l
T
Because
R^
belongs to R^ .
by Corollary 10.8, T^ =
induces
- T. for
an isomor
If
R^ e
S
each i andthat
i
each R^is in
S* , then
is a unit of
T^
Thus
zl?u. is a unit of 1 l
nilpotent.
T ;
simultaneously.
T^[Z] for each i .
u^
of
the identity element of
nilpotent for each i .
z?t. where 1 i ’ is
S and
.It follows that
+ t^ , where
is
is
.
are in
On the other hand, if
^(e^X) = u^X
M
P
T = T^©. . .©Tn , where
for eachi , we conclude that *
t^ e T^[Z]
|<J>|
T[Z]
T and
of R^[Z]
R - T .
is elementary.
1 = |<J>* | =
Since the idempotents of
T(e^)
<J>
R = R^@...©Rn , where each
a specified class.
Local Rings » Wiley (Interscience), New
K * Nicholson, 137-138.
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(x,y)n = (xn .yn )
Ohm and P. Vicknair, Monoid rings as valuation Algebra 11(1983), 1355-1368.
114. M. J. O'Malley, Isomorphic power series Pacific J. Math. 41(1972), 503-512.
rings,
115. M. Orzech, Onto endomorphisms are isomorphisms, Amer. Math. Monthly 78(1971), 357-362. 116. T. Parker, The semigroup ring, Florida State University Dissertation, 1973. 117. T. Parker and R. Gilmer, Nilpotent elements of commutative semigroup rings, Mich. Math. J. 22(1975), 97-108. 118. M. M. Parmenter, Isomorphic group rings, Canad. Math. Bull. 18(1975), 567-576. 119. ,Coefficient rings of isomorphic group rings, B o l . Soc. Bras. Mat. 7(1976), 59-63. 120. M. M. Parmenter and S. Sehgal, Uniqueness of the coefficient ring in some group rings, Canad. Math. Bull. 16(1973), 551-555. Wiley
121. D. S. Passman, Algebraic Structure of Group Rings, (Interscience), New York, 1977.
122. L. Redei, The theory of finitely generated commu tative semigroups, Pergamon, Oxford-Edinburgh-New York, 1965. rings.
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,Topics in Group Rings, Marcel
group
group rings, Dekker, New
126. T. S. Shores, On generalized valuation rings, Mich. Math. J. 21(1974), 405-409. Math.
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128. M. L. Teply, Semiprime semigroup rings and a problem of J. Weissglass, Glasgow Math. J. 21(1980), 131-134. 129. M. L. Teply, E. G. Turman, and A. Quesada, On semi simple semigroup rings, Proc. Amer. Math. Soc. 79(1980), 157-163. group
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131. J. Weissglass, Amer. Soc. 25(1970),
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II, Van Nostrand,
TOPIC INDEX
The following abbreviations are used in some of the listings in the index:
cong. for congruence, dom. for domain,
elt. for element, g p . for group, id. for ideal, mon.
for
monoid, and smgp. for semigroup. m a p . .75, 89, 102, 148, 279 Automorphism of a mon. ring.. 303, 306, 309
A Adjunction of an identity elt. canonical to a ring..104, 111 to a smgp...2, 275, 328, 331 Affine domain..289, 296, 299 Algebraic retract..147 Almost integral ring elt... 152 sm gp . e l t ...151 Annihilator of a ring elt... 94 Aperiodic smgp. elt.,.10, 326 Archimedean component of a smgp...236, 323, 326 decomposition of a smgp... 236, 322, 325 Ascending chain condition (a.c.c.) on cong...30, 39, 271, 272 on cyclic submonoids..181, 187, 191 on cyclic subgroups..181 on principal smgp. ideals.. 57 on principal ring ideals.. 171, 191 on smgp. ideals..40, 152 on subgroups..271 on submonoids..13 Associate elements of a mon... 52, 57 Augmentation ideal..75, 164, 277, 279, 338
B Basis of a free gp...l92, 309
219,
C Cancellation law for ring ideals..169 Cancellative semigroup elt...4 Canonical adjunction of identity elt. to a rin g..104, 111 form of a smgp. ring elt...
68 representation of a cong... 47 Center of a valuation on a g p ...221, 223 overring..194 Characteristic of a field..136, 254, 261, 279, 352 a ring. .98, 114, 118 , 125,' 136, 302 an integral dom...90 Chinese Remainder Theorem.. 243
363
364 Class g p . of a Dedekind dom. 343 Cohen's Structure Theorem.. 270 Comaximal ring ideals..164, 264 Complete integral closure.. 151, 153 Component homogeneous, of a graded r in g..85, 102 primary, of a group..100, 352 of a ring ideal..261, 266 Congruence(s) a.c.c. o n . .30, 271, 272 asymptotic equivalence..34, 97, 228, 320 cancellative..31, 44, 136, 227, 272, 324 canonical representation.. 47 commutative..318 d.c.c. o n . .271, 272 definition o f . .28 A . .35, 86 factor smgp. of..28 finitely generated..30, 72 generated by a set.. 29, 318 identity..29 irreducible..46 kernel id. of..79, 90, 99 lattice o f ..29, 71 on a g p ...33 p— equivalence..34, 97, 228, 320 primary..44 p r i me ..44 Ree s..36, 27 2 restricted to a subsemi group ..44 universal..29, 42 Content of a polynomial..293 smgp. ring elt...96, 113, 121, 131 Cosets of a subgroup..4 Cyclotomic field extension.. 352 D Dedekind-Mertens Lemma..96 Defining family for a Krull d om ...197 Degree of
a smgp. ring elt...84, 129 an elementary isomorphism.. 339 Descending chain condition (d.c.c.) on congruences..271, 272 on smgp. ideals..272 on subgroups..271 Dimension of a ring..226, 227, 288, 290, 328, 330, 343, 347 Direct sum of groups complete..25 external weak..26, 223 internal weak..77, 85, 100, 124, 183, 213, 272, 286, 321, 346, 351 into primary components.. 100 of m od ules..148, 149 of rings..100, 117, 123, 130, 142, 249, 250, 264, 266, 269, 308, 316, 338, 343, 347, 351 of semigroups external..18, 66, 180 weak internal..11, 61 Distributive law for ring ideals..243 Divisible hull of a group.. 163, 241 Divisor class group of a domain..209, 220, 223 of a m o n o i d ..219 Divisor class monoid of a d omain..208 of a m o n o i d ..217 Divisorial fractional ideal of a domain..208 of a m o n o i d ..215 E Element of a group divisible by an integer.. o f ^ y p e (0,0, ... ,0) . .180, 187 191 Element of a ring almost integral..152 annihilator of..94 idempotent..113, 308, 339 integral..152 nilpotent..117, 303, 305, 338
365 prime..183 u ni t..117, 303, 305 zero divisor..82, 89 Element of a smgp. or monoid almost integral..151 aperiodic..10, 326 associate..52,57 cancellative..4 divisor o f . .52 factor o f . .52 idempotent..11, 135, 231, 237 identity..2 index o f ..10, 229 integral..151 invertible..4, 52 irreducible..52, 56 multiple o f . .52 order o f ..10, 229 period o f ..10 periodic..10, 120, 326 p rime..52, 56 u ni t..52 Element of a smgp. ring or mon. ring canonical form of..68 content of..68, 96, 113, 121, 131 degree o f ..84, 129 G— component..134 I— component..134 idempotent..113, 135 invertible..120, 129, 131, 140 homogeneous..174 homogeneous content of..174 homogeneously primitive.. 174 mo ni c..84 monomial..89, 92, 113, 129, 172, 178 nilpotent..97 order o f ..84, 129 support of..68, 91, 113, 131, 138, 242 supporting smgp. of..68 zero divisor..82, 304 Element prime to a ring sub set . .17 3 Elementary isomorphism..338 degree o f ..339 Endomorphism injective..304, 306, 315 surjective..303, 306, 315 Exponent of a gp...267
F Factor smgp. of a congruence.. 28 Field algebraically closed..184 characteristic of..136, 254, 261, 279, 352 Fractional ideal of a domain divisorial..208 of a monoid..215 divisorial..215 principal..215 Free basis of a group..192, 219, 309 of a module..148, 150, 341 G Galois group..185 Gcd of smgp. elements..52, 54, 59, 175 Generating set for a smgp... 3 for a smgp. ideal..3 Gib of a family of congruences 29, 43, 72 Graded ring..85 homogeneous elt. of..85 Grading on a ring..85, 86, 174 Group a.c.c. on cyclic subgroups..181 cyclic submonoids..181 subgroups..271 cyclic..17, 168 d.c.c. on subgroups..271 divis ible..163 divisible hull of..163, 241 exponent o f ..267 finitely generated..41 free..192, 226 free basis of..192, 309 locally cyclic..16 mi x e d ..6 Noetherian..41 of divisor classes of a d o m . ..209, 220, 223 of a m o n ... 219 of invertible elts. of a mon. 120 p— quasicyclic..255
366 primary component of..280, 281 projection map on..192 quotient g p . of a cancella tive smgp...6, 93, 227, 237, 241, 288, 290 subgroups linearly ordered 256 torsion..6, 165 torsion— free..6, 165, 181 torsion— free rank of..165, 226, 280, 290, 350 torsion subgroup..121, 282 Group ring..88, 92, 120, 123, 127, 128 integral..346, 352 isomorphic..351 rational..352 H Height of a prime ideal..289, 293, 294 Homogeneous component of a graded ring..85, 102 an element..86 content of a mon. ring elt. 174 decomposition of an elt...86 elt. of a graded ri ng ..85 smg p. ring..174 Homogeneously primitive..174 Homomorphism of a smgp. ring 69 I Ideal of a ring contracted..149, 162, 243, 280, 330 finitely generated..72 idempotent..116, 121 invertible..243 maximal..330 minimal prime..212, 347, 348 multiplication..249 n i l . .116 primary..87, 212 p r i me ..83, 330 Ideal (s) of a smgp. or monoid 3, 36 d.c.c. on..272
prime..3, 40, 120, 136, 227, 273, 326 pr oper..3 radical of a n . .3, 43, 44 , 233, 323 Ideal of a smgp. ring augmentation..75, 164, 277, 279, 338 idempotent..320 kernel, of a cong...70, 90, 99, 271 Idempotent elt. of a ring..113, 308, 339 smgp...11, 135, 231, 237 smgp. ring..113, 135 Identity elt. of a mon...2 Imbedding a cancellative smgp. in a g p . 5, 7 smgp. in a monoid..2 Index of a smgp. elt...10, 229 Injective endomorphism..304, 306, 315 Integer regular on a ring..110 Integral closure of a mon...151 dependence in rings..166, 227, 291, 299 extension of a ring..125, 280 ring e l t ...152 smgp. element..151 Integral domain (or domain).. 82, 96 a.c.c.p. in..149, 171, 177, 188, 191 almost Dedekind..169, 189, 277, 285 a tomic..189 Bezout..162, 167, 240 characteristic of..90 complete integral closure.. 153 completely integrally closed 153, 209 Dedekind..162, 168, 240, 263 343 Euclidean..168 factorial..149, 171, 187, 196 GCD— domain..149, 171, 176 integral closure..158 integrally closed..147, 162 185, 206, 268 Krull domain..190, 198, 209 Noetherian..168, 206, 290 of finite character, etc. 206
367 II—domain . .207 PID..162, 168, 240, 267 Prufer..162, 167, 240, 302 valuation..169, 190, 209, 270 Integral group ring..346, 352 Irreducible monoid elt...52, 56 Isomorphism of group rings..351 monoid rings..337, 351 J Jacobian conjecture..315 Jacobson radical of a monoid ring..129, 133, 140 ri ng ..347 Jaffard’s Special Chain Theorem..288 K Kernel id. of a cong...70, 90, 99, 271 Krull domain..190, 198 defining family for..197 Krull Intersection Theorem 122, 128 L Lattice of congruences on a smgp...29, 38, 71 ideals of a ring..71 subgroups of a gp...7 5 Lem of ring elements..173 smgp. elements..53, 54, 59, 172 Lifting idempotents..116 Lub of a family of congruences 29, 35, 37, 72 Lying—over Theorem..280 Lying—under Theorem..186, 268 M McCoy's Theorem..84
Minimal prime of a ring ideal 186 Monic smgp. ring elt...84 Mono id ..2, 93, 119 a.c.c. on cyclic submonoids 181, 187, 191 aperiodic..135 cancellative..42, 52, 129, 148, 153, 162, 171, 212, 215, 218, 240, 242, 253, 289, 350 completely integrally closed 151, 214, 217 cyclic..167 divisor class mon...208, 217 factorial..57, 181, 187 finitely definable..76 finitely generated..168, 204, 289. finitely presented..76 free . .65, 77, 78 GCD-monoid..55, 171 integral closure..151, 156 integrally closed..151, 166, 206 Krull..190, 198 Noetherian..68, 76 numerical..12, 34 p e ri odi c ..260, 273 primitive..12 Prufer..166 quotient mon...6, 31 rank o f . .15 torsion— free..129, 153, 162, 171, 212, 218, 240, 289, 350 unique factorization..57 valuation..199 Monoid of divisors of a domain..208 of a mono id ..219 Monoid ring..67, 125, 129 arithmetical..167, 247, 253, 259, 261 Artinian..271, 274 as a graded ring..174 automorphisms of..303, 306, 309, 339 Bezout..167, 247, 253, 259, 263 chained..254, 256 completely integrally closed 155 definition of..67 dimension of..288 factorial..171, 187 GCD— domain..171, 176 general ZPI— ring..250, 253, 265
368
integrally closed..158, 159 isomorphism of..337, 351 elementary..338 degree o f ..339 Jacobson radical of..129, 133, 140 Krull domain..190 local..255 locally Noetherian..271, 280, 283 multiplication ring..250 nilradical o f . .97, 130, 140 Noetherian..75, 266, 271 P— ring..247 PIR..168, 250, 253, 265, 267 Priifer ring..167, 247 , 263 quasi— local..253, 255 reduced..138 RM— ri ng ..271, 277 satisfying a .c .c . p ...191, 271 semi—quasi— local..270 semilocal..270 trivial units of..312, 315, 347, 351 unit o f ..129, 131, 140, 167, 351 von Neumann regular..226, 228, 238, 259 Monomial..89, 92, 113, 129, 172, 178 p u r e ..205, 292 Multiplicative system in a ring..111 N Natural topology on a local ring..284 Nilpotent elt. of a ring..117, 303, 305, 338 smgp. rin g..97 Nilradical of a monoid ring..97, 130, 140 ring..89, 97, 131, 347 Noether Normalization Lemma.. 302 0 Order of nilpotency of a ring id.. 304, 311 a smgp. element..10, 229 a smgp. ring elt...84, 129 334
Order relation cardinal..26, 192, 203 lexicographic..26 partial..22 positive cone..24 reverse lexicographic..26 total..22, 153, 156 Orthogonal idempotents..141, 339 P Partial order definition o f ..22 on congruences..29 Period of a smgp. elt...10, 120 Periodic smgp. elt...10 Polynomial ring..64, 65, 78, 154, 159, 172, 226, 246, 288, 293, 303, 308, 317, 338, 351 Positive co n e ..24 Positive subset..24 Power series ring..21, 315, 336 Primary component of a group..100, 352 a ring ideal..261, 266 Primary Decomposition Theorem for congruences..48 for ring ideals..122 Prime element of a domain..183 a smg p... 52, 56 Prime subring..305 Projection map on a gp...l92 Pure difference binomial..78 monom ia l..205, 292 subgroup of a gp...l83
Q Quotient gp. of a cancellative smgp.. , 93, 227, 237, 241, 288, 290 monoid of a smgp...6, 31 overring of a domain..209 ring of a ring..Ill, 241, 243, 246, 253, 261
369 R
quasi— local..253 , 254 , 282 , 347 reduced..97, 132, 139, 156, 226, 312, 316, 320 R e e s ..344 restricted minimum condition 276, 286 S— invariant..345 second chain condition..286 semiprime..112 semisimple..129 special PIR (SPIR)..88, 249 , 264 special primary ring..249 von Neumann regular..113, 226, 228, 242, 259, 277, 328 Z— invariant..345 zero— dimensional..226, 271, 274 ZPI— ring..251 Root of unity..267, 332
Radical of a ring ideal..88 smgp. ideal..3, 43, 44, 233, 323 Rank of a m o n ... 15 Rank (torsion— free) of a gp... 165, 181, 226 Rational g p . ring..352 Relation asymmetric..22 compatible with smgp. opera tion ..22 partial order..22 total order..22 Ring arithmetical..240, 243, 251, 253, 257, 264 Artinian..257, 271, 286 Bezout..240, 244, 253 Boolean..113, 239 chain condition for primes.. 286 S chained, valuation..243, 253, 256, 270 Saturated chain condition..289 FMR-ring..103, 118, 139 general ZPI— ri ng ..240, 249, Semigroup..2 251, 253, 264 abelian..2 aperiodic..10, 119 graded by a smgp...85 Hilbert..102, 118, 139 Archimedean..233 cancellative..4, 7, 22, 31, idempotent..318 35, 44, 82, 106, 119, 129, idecomposable..113, 124, 126, 132, 312, 347, 352 236, 322 integrally closed..258 commutative..2 invariant..335 cyclic ..9, 114 local..256, 283 factor smgp. of a cong...28, 318 locally Cohen-Macaulay..286 finitely generated..68, 275 free of asymptotic torsion.. Gorenstein..286 Noetherian..277, 286 34, 105, 228, 236, 323 homomorphis.m of a.. 14 regular. .286 multiplication..240, 251 monogenic..9 Noetherian..30, 39 n— invariant..335 ni l. .98 numerical..34, 333 p— torsion— free..34, 99, 228, Noetherian..68, 75, 122,154, 249, 250, 255, 264, 302, 324 306, 309, 337, 343 partially ordered..22 periodic..7, 10, 227, 318, P—r i ng ..241, 247 PIR. .240, 253, 266 320, 328 polynomial..64, 65, 78, 154, separative..106 159, 172, 226, 246, 288, supporting..68 torsion— free..7, 22, 33, 35, 293, 303, 308, 317, 338, 82, 104, 129 351 totally ordered..22, 84 power invariant..336 Semigroup ring power series..21, 315, 336 Artinian..27 5 Prufer..240, 244, 251, 258
370 split by idempotents..141, 308 trivial..129, 133, 141, 144, 312, 315, 347, 352 ri ng ..117, 303, 305
definition o f ..64 homomorphism on..69 integral domain..82 isomorphism of..317, 331, 333 nilradical of..97, 320 Noetherian..77, 275 reduced..97, 320, 325 zero divisor..82, 304 Semilattice..231, 234, 239 Semilattice of subsemigroups.. 233 Shortest primary representation 264 Special chain of primes..288, 298 Split extension of subgroups.. 314 Subgroup maximal containing 0..4 of a smgp...2, 32, 93, 231 pure...183 pure subgroup generated by a s et ..183 torsion subgroup of a g p ... 121, 282 Subisomorphic rings..340, 342 Subsemigroup..2 Support of a smgp. ring elt... 68, 242 Surjective endomorphism..303, 306, 315 T
V v— operation on a domain..208 m on oi d..217 Valuation domain..169 center on a subring..194 discrete..190 essential..194 rank— one..190 Valuation monoid..200, 202,
220 essential..201 Valuation on a field..194, 200, 211 group..155, 220, 223 center..221, 223 discrete..190 essential..201 family of finite character 190 nontrivial..155 normed..203 rank o f ..190 trivial..155 Valuation overring..190 Value mon. of a g p . valuation 155, 190
Torsion— free rank of a g p ... 165, 181, 290, 350 Z Torsion subgroup of a gp.,.121, 282 Total quotient ring..149 Zero divisors of a Transcendence degree..289 ri n g ..82 Trivial units of a mon. ring.. s mg p. ring..82, 304 129, 133, 141, 144, 312, 315, 347, 352 Type of a g p . elt...180, 191, 212 U Unital extension of a ring..67, 68, 91, 93, 98, 111, 119 Units of a monoid ring..129, 131, 140, 169, 351 of finite order..143
INDEX OF MAIN NOTATION
SYMBOL S^
PAGE
MEANING
2
Monoid obtained element to S
by
adjoining an identity
A + B
2
Sum of subsets
3
Subsemigroup generated by
rad(I)
3
Radical of the ideal
Z
5
The integers
Zq
5
The set of nonnegative integers
c
7
Set inclusion
*
69
The kernel of
72
least upper bound of congruences
72
greatest lower bound of congruences
77
Direct sum of
90
Characteristic of
Zn Char D PID
divides
S
p and
associated with
p
[p]°
b
for
A S
over
R
f
f *
n
copies of
Zq
D
165
Principal ideal domain
165
The torsion— free rank of
(g)
180
Cyclic subgroup generated
zw
185
Direct sum of
208
Divisorial ideal associated with
div (A)
208
The divisor class of
PCD)
208
The set of divisor classes
of
P(D)
209
Principal divisor classes
of
r0 (M)
Fv
s
w
M by
copies of
g Z F
A D D
373 SYMBOL
PAGE
MEANING
C (D)
209
Divisor class monoid of
I v
215
Divisorial ideal
of
div(I)
217
Divisor class of
I
PCS)
217
Divisors classes
of
PC S)
217
Principal divisor classes of
C(S)
217
Divisor class monoid of
SPIR
249
Special principal ideal ring
RS
317
Semigroup ring of noncommutative
2ir isi
317
Element of
deg
where
379
RESULT
PAGE
DESCRIPTION
17.3
227
dim R[S] = dim R
17.4
228
Equivalent conditions for (von Neumann) regular
17.9
234
Archimedean decomposition
17.10
236
The Archimedean components are cancel lative iff S is free of asymptotic torsion
18.5
242
If R[S] regular
18.6
244
R is arithmetical iff each chained
18.7
246
R regular implies ring
18.9
248
Equivalent conditions for R[S]to be a Priifer ring, a Bezout ring, or an arithmetical ring
18.10
250
The conditions PIR ,general ZPI— ring, and multiplication ring in R[S]
19.1
254
R[U] is quasi-local iff R is quasi— local, Char(R/M) = p * 0 U is a p— group
isa
iff
S
is periodic R[S] of
P— ring, then
R [X ]
to
be
S
R RM
is a
is is Bezout
and
19.4
256
Conditions
19. 7
259
Conditions for R [S ] to be arithmeti cal, where S is neither torsion— free nor periodic
19.13
265
R[S]
as a
ZPI— ring
19.16
267
D[G]
as a
PID
20. 5
273
If S satisfies a.c.c. and d.c.c. on congruences, then S is finite
20 . 6
274
R[S] and
S
for
R[U]
to be
is Artinian iff is finite
20.11
280
Necessary conditions for locally Noetherian
20.14
284
Converse of Theorem 20.11
21.4
290
dim R[S] = dim R[G] lative
if
chained
R is Artinian R[G]
S
is
to be
cancel
380 RESULT
PAGE
DESCRIPTION
21.9
298
22.1
303
Determination of the of R[X]
22.3
306
R— automorphisms of
R[X,X *]
22.4
309
R— automorphisms of
R[Zn ]
23. 12
329
R [S* ] is the maximal regular subring of R[S]
23. 16
331
F[S] « K[T] periodic
23.17
333
F - K if aperiodic
24.2
338
Reduction to the case of elementary isomorphisms
24.6
342
R[Z] - T [ Z ] implies subisomorphic
24.9
346
R[ Z]
24.11
347
Sufficient conditions for isomorphism of R[Z] and T[Z] to imply that of R and T
Special Chain Theorem holds in R[S] for S torsion— free and finitely generated
implies
R— automorphisms
F
F [S] = K[T]
= T [ Z ] iff
R
R[Zn ]
= K and
if S S
and
is
is not
T are
= T[Zn ]
