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EPlMORPHISMS AND SUBALGEBRAS 0F

FiNlTELY GENERATED ALGEBRAS

Thesis for the Qegree of Ph. D.
MICHIGAEV’ STATE UNSVERSWY
DEAN EDWARD SANDERS
1972

 
 
 
   

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EPIMORPHISMS AND SUBALGEBRAS OF
FINITELY GENERATED ALGEBRAS

presented by

Dean Edward Sanders

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ABSTRACT

EPIMORPHISMS AND SUBALGEBRAS OF
FINITELY GENERATED ALGEBRAS

BY

Dean Edward Sanders

In any category a morphism f is an epimorphism if for
any two morphisms g and h the equality of g o f and
h 0 f implies the equality of g and h. In the category
of rings, f: R 4 S is an epimorphism if and only if for
every element 5 of S there exist two finite sets, {Si}
and {st}, of elements of S and a finite set, {rij}, of

3

elements of R such that: 1. s = 23s; rij s; , 2. for
13'
II

each i, Z rijsj is in f(R), and 3. for each j, §s£ rij
is in f(R). Let R be a commutative ring and S a faith-
ful R-algebra. If the inclusion map from R to S is an
epimorphism, then S ‘will be called an epimorphic extensigg
of R.

Let R be a commutative ring, S a faithful R—algebra
and M a left S-module which is finitely generated as a
left R-module (under the induced structure). The above char-
acterization of an epimorphism is used to prove that if S
is an epimorphic extension of R, then for any element m of

Dd, S-m = R-m . This theorem has a chain of consequences.

Let R c S c A be rings with R commutative, A an R-

Dean Edward Sanders

algebra which is finitely generated as an R-module, and S

a subalgebra of A. It follows from the preceding theorem
that S is an epimorphic extension of R if and only if

S = R. This is used to prove that if S is a separable R-
algebra then S is finitely generated as an R-module. Fin-
ally, this result is used to prove that if S is an inertial
subalgebra of A and A is commutative with Jacobson radi—
cal N, then S = n{B|B is a subalgebra of A and A = B4-N];
hence, S is unique.

The condition that A be finitely generated as an R-
module implies that S is an integral extension of R. and
there exists a positive integer n such that every element
of S satisfies a monic polynomial of degree n having co—
efficients in R. Chapter III investigates integral epimor-
phic extensions. Let R be commutative and let S be a
faithful R-algebra. First it is shown that if R is Noeth-
erian or is an integrally closed domain,then S is an inte-
gral epimorphic extension of R if and only if S = R.
Finally, two examples are given of a domain R and a faith—
ful integral epimorphic extension 8 of R with R # S.

In each example R and S are subrings of a discrete group
ring where the group is a non-archimedean ordered group. In
the second example every element of S satisfies a monic

polynomial of degree 2 having coefficients in R.

EPIMORPHISMS AND SUBALGEBRAS OF
FINITELY GENERATED ALGEBRAS

BY

Dean Edward Sanders

A THESIS

submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Mathematics

1972

ACKNOWLEDGMENTS

I wish to thank my advisor Dr. E. C. Ingraham for sug-
gesting this line of research and for offering helpful com-
ments and suggestions. His patience with and continued
interest in both student and problem are greatly appreciated.
Special thanks go to my wife Bonnie for her continued en-
couragement and understanding. Finally I wish to thank Bill
and Jean Horiszny for the generous use of their "magic house"

as a peaceful place in which to work.

ii

TABLE OF CONTENTS

Page

INTRODUCTION . . . . . . . . . . . . . . . . . . . . 1
Chapter

I. EPIMORPHISMS. . . . . . . . . . . . . . .-. 4

1. Preliminaries . . . . . . . . . . . . . 4

2. Rings of Quotients. . . . . . . . . . . 5

3. Epimormphisms of Rings. . . . . . . . . 7

4. Epimorphisms over Commutative Rings . . 12

II. SUBALGEBRAS OF FINITELY GENERATED ALGEBRAS. 18

1. Central Epimorphisms. . . . . . . . . . 18
2. Separable Algebras. . . . . . . . . . . 22
3. Epimorphisms and Separable Algebras . . 24
4. Inertial Subalgebras. . . . . . . . . . 28
III. INTEGRAL EPIMORPHIC EXTENSIONS. . . . . . . 31
1. Integral Extensions . . . . . . . . . . 31
2. Integrally Saturated Rings. . . . . . . 32
3. Examples of Integral Epimorphic
Extensions. . . . . . . . . . . . . . . 37
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . 45

iii

I NTRODUCT I ON

Let R be a commutative ring, S an R-algebra and
8° the opposite algebra of 8. Let H: S ®R 5° 4 S be
the surjection defined by “(s O s’) = 33’. The opera-
tion (3 ® s’)°s” = ss”s’ gives S the structure of a
left S ®R S°-module. If the sequence 0 4 ker u 4 S ®R8°
u

4 S 4 O splits as left S ® S°-modules, then S is

R
called a separable R-algebra. Let A be an R-algebra
which is finitely generated as an R-module, and let N de—
note the Jacobson radical of A. An inertial subalgebra of
A is a separable subalgebra S such that A = S + N.
In 1951 G. Azumaya [A], using a different terminology,
investigated existence and uniqueness questions for iner-
tial subalgebras which are finitely generated as R-modules.
Since that time it has remained an open question as to wheth-
er or not an inertial subalgebra is necessarily finitely
generated. This thesis answers that question in the affirm-
ative. The tool that is used is the concept of an epimor-
phism in the category of rings.

In any category a morphism f is called an epimorphism
if for any two morphisms g and h the equality of g o f
and h o f implies the equality of g and h. Let R
and S be rings, with identity. A ring homomorphism
f: R 4 S is an epimorphism if and only if the surjection
I

u: S ®R S 4 S defined by “(5 ® 5’) = ss is injective.

This characterization demonstrates the connection between

1

epimorphisms and separable algebras. If f: R 4 S is a
ring epimorphism, S ‘will be called an epimorphic extggr
gigg of R.

Chapter I is a brief investigation of epimorphisms of
rings. Theorem 1.1 gives 10 characterizations of an epi-
morphism in the category of rings. The remainder of the
chapter is a collection of basic results regarding epimor-
phisms.

Chapter II is the focal point of the thesis. Let R
be commutative, A a faithful R-algebra which is finitely
generated as an R-module, and S a subalgebra of A. First,
it is shown that if S is an epimorphic extension of R
then S = R. Second, this result is used to show that if
S is a separable subalgebra of A then S is finitely
generated as an R-module. Finally, the second result is
used to prove that if S .is an inertial subalgebra of A
and A is commutative, then S = n [BIB is a subalgebra
and A = B + N}. A uniqueness statement follows immediately.

Returning to the setting of chapter II, the condition
that A be a finitely generated R-module forces the sub-
algebra S to be an integral extension of R. Moreover,
if A is generated as a module over R by n elements,
then every element of S satisfies a monic polynomial in
R[X] of degree n. Chapter III investigates integral epi-
morphic extensions. First it is shown that if R is an
integral domain which is integrally closed in its total quo-

tient field or is a commutative Noetherian ring and if S

is a faithful integral epimorphic extension of R, then

S = R. The thesis concludes with two examples of non-
trivial integral epimorphic extensions of domains. In the
second example every element of the extension satisfies a
manic polynomial of degree 2 with coefficients in the ground

ring.

Chapter I

EPIMORPHISMS

1. Preliminaries
Throughout this thesis all rings have identity, and all
ring homomorphisms map identity elements to identity ele-
ments. A subring will contain the identity element of the
larger ring. The identity map on a set E will be denot-
ed by IdE or, when there is no danger of confusion, by
Id. Mappings defined on a tensor product will be defined
on monomials and are to be extended linearly to all of the
tensor product. The unadorned symbol Q 'will always mean
®R.
If f: R 4 S is a ring homomorphism, then f induces
a left (resp. right) R-module structure on S defined by
r ° 3 = f(r)s (resp. s - r = sf(r)) for every r in R
and every 5 in S. The ring S will be said to have a
module property if it has that property as a left R-module
under the structure induced by f. The subring f(R) of
S will often be denoted'by R-l. If S is faithful as
a left R-module, then f is a bijection and R a f(R): in
this case R ‘will be considered as a subring of S. A ring
S together with a ring homomorphism from a commutative ring
R into the center of S will be called an algebra over R.

For the most part, background material will be intro-

duced in an expository fashion. In each case a reference

containing a detailed discussion will be cited.
4

2. Rings of Quotients

This section contains a brief discussion of rings of
quotients and their connection with local rings and flat
modules.

Let R be a commutative ring. A subset W of R is
called multiplicatively closed if: 1) 0 g W, 2) l €‘W,
3) if W1 and *w2 are in W so is their product. For
a multiplicatively closed subset W of R and a right R-
module M, define an equivalence relation ~ on M.x'W by
(m1,wi) ~.(m2,w2) if there exists w E'W such that

(ml'w2 - m2w1)w = 0. Let m/w denote the equivalence class

of (m;w). The set of equivalence classes forms a right R~
module under m m'w + m*w
_+_2_=__1__2____2__l.
w1 W2 w1W2
and
m1 mlr
.7“ ”T
1 1

This module, denoted by MW, is called the module of guo-
tients of M with respect to W. If M is a ring, then
MW is a ring under the multiplication

_l‘_“_2_ "‘lmz
W1 W

 

A)
P?
t

The ring RW is called a ring of quotients of R, and
{r/l I r 6 R] is a subring of RW which is isomorphic to
R provided W contains no zero—divisors of R. If ‘W is
the set of all elements which are not zero-divisors of R,

then RW is called the total ring of quotients of R.

Finally, if R is a domain, the total ring of quotients of
R is a field containing R.

It is easy to check that if M is a right R-module
(resp. R-algebra), the map from M ®R Rw to Mw defined
by m ®;§'4 %f- is an R-module (resp. R-algebra) isomorphism.
Whenever it is needed, this isomorphism will be employed

without further comment.

Definition - Let R be any ring. A (left) R-module M
is called flag if for any exact sequence 0 4 N E N’ of
(right) R-modules the sequence 0 4 N o M {933 N’ G M is
exact.

Let R be a commutative ring and W a multiplicatively
closed subset of R. Let 0 4 N g N’ be an exact sequence
of (right) R-modules. It is easily seen that O 4 NW f-VNW’
is exact, where fwe) = _f_‘_va)_)_ . Because of the isomorphism
between the right R-modules MW and M ®R Rw, the exact—
ness of O 4 wa-y Né is equivalent to the exactness of
O 4 N ® Rw £§E§ N” 8 RW” Thus, any ring of quotients of
R is a flat R-module.

The complement of any prime ideal p of R is a mul-
tiplicatively closed subset of R. The corresponding mod-
ule of quotients is denoted by Mp rather than MR-p’ It
is easily seen that the ring Rp has pRp = f§|p e p,r(;R-p}
as a unique maximal ideal. A commutative ring, not necessar-
ily Noetherian, having a unique maximal ideal will be called

a local;rinq. The ring Rb is called the localization of

Rat p.

3. Epimorphisms of Rings

In this section we define an epimorphism and give sev-

eral equivalent conditions in the category of rings.

Definition - In any category a morphism f is called an
epimorphism if for any two morphisms g and h the equality
of g o f and h 0 f implies the equality of g and h.
The following theorem gives several characterizations
of epimorphisms in the category of rings. Condition 2 was
proven by Mazet [11], conditions 3-7 were proven by Roby
[19], and conditions 3,5,8,9,10 were proven by Storrer [23].
Proof of the equivalence of 1 through 6 is given. Since 7
through 10 are not used significantly in this thesis, the
interested reader is referred to the literature for proofs.
The following notation is used in 8,9, and 10. If M
is a left (resp. right) S-module and f: R 4 S is a ring
homomorphism, then the underlying abelian group of M can
be given the structure of a left (resp. right) R-module by
defining r . m = f(r)m (resp. m - r = mf(r)). This R-

module is denoted by M’

Theorem 1.1 - Let R and S be rings, not necessarily com-
mutative, and let f: R 4 S be a ring homomorphism. The
following statements are equivalent:

1) f: R 4 S is an epimorphism.

2) For every 5 6 S there exist two finite sets of elements

Of S. [Si] and {8%}, and a set of elements of R,[rij],sucn

that :

i) s = Zfsi r.. s. ,

ii) for each i. Fjrij s? is in f(R),
iii) for each j, £)si rij is in f(R).
i

3) For every 5 E S the relation 3 G l - 1 a s = 0 holds

in S ®R S.

4) The surjection u: S ®R S 4 S defined by “(s @ s’)==
38’ is injective.

5) The injection p1: S 4 S ®R 8 defined by p1(s) = 5 ® 1
(resp. p2: S 4 S ®R S defined by p2(s) = l G s) is
surjective.

S S

 

 

7) TR(S), the tensor algebra on R of the R-module S. is
commutative.
8) For every (right) S-module M, the S-modules M and
MI ®RS are isomorphic.
9) For every pair of S-modules M and N
a, I I
HomS(M,N) _ HOmR(M ,N ).
JIM For every right S-module M and every left S-module

., I I
N, M®SN.--M®RN.

Proof:
3 a 2. This implication relies on the following lemma

[3, chap. 1, § 2, n°11, lemma 10] .

Lemma - Let A be a ring with l, E a right A-module, and
F a left A-module. Let {eili E I] be a set of generators
of E, [lej 6 J} a set of generators of F, and

[fi Ii 9 I} a set of elements of F having finite support.

In order that Z) e. S f. = O in E S F, it is neces-
iEI i 1 A

sary and sufficient that there exist a set of elements of

A, {aij|(i,j) e I x J], having finite support such that

i) for each i E I, f. = Z) a..x. and
1 jEJ 13 3
ii) for each j e J, Z) e. a.. = O.
ieI 1 13

£00f0f3ag.

Let s 6 S, then 1 S s - s 8 l = O in S Q S. If
[si|i e I] is a set of generators for the right R-module
S, then there exists a set of elements of R, [ri|i E I],
having finite support such that s =:Z}siri. Hence,
183-23i Gri-l= O in SQS. Assume O£I and
set I’ i I U {0}. Applying the lemma we obtain a set of
elements of R, [rij|(i,j) E I’ x I], having finite sup-

port such that
i) for each 16 I, r.-l = Drij s. and
3

ii) for each j e I, l.'r . - Z)s.r.. = O.
0.] i#0 1 13

This implies that:
i) s = Z)s.r. = Z33. (Z)r. =Z)s. r.
i j

1 1 i 1 ij Sj) i j i ij5 j
11) for each 1, §2rij sj == rrl e f(R), and
111) for each 3, stirij == l-rOj 6 f(R).

2 s 1. Let g and h be two ring homomorphisms from S

to a ring T such that g o f = h o f. Then for r e R,

I

s .s” e S. 9(8’)h(r-s”) = 9(s’)h(f(r)s”) = 9(8’)h(f(r))h(s”)

9(8’)9(f(r))h(8”) = 9(8’f(r))h(s”) = 9(8’r)h(8”). Let

s 6 S and assume 3 has the indicated representation.

lO

. 7 F f R th t 3 .. 7 = h I :
Then Z‘rijsJ , ( ) so a g(é r1383) (gr ij 8;)
similiarly, 9C? 5; rij) = hXZIS’r rij). These relationships
1
imply that g(s) = g(Z s; (Pgri j 83))
i

I - II
§g(si)90§l rij 83-)

= 23 g(S’)h(Z rij 8’31)

i i j

=53jglsi )h(rijs§)
=iZ“jg(si rij)h(sg)
= j Tg(Z‘sirij)h(s”)
= ? bu: sirij)h(s”)

h s’r..si) = h s .
10.33%“) H

Therefore g = h.

l a 3. The injections p1 and p2 of S into S G S de-
fined in 5 are ring homomorphisms such that p1 o f =
pzof.

Since f is an epimorphism, p1 = p2.

3 a 4. The kernel of u is precisely the two-sided ideal
of S a 8 generated by all elements of the form 183 - 88> 1.
Thus u is injective iff ker u = 0 iff for every 3 6 S,

l p s = s 8 l in S S S.

4 s S. The composition u 0 p1, (resp. u 0 p2) is the
identity map on S. Therefore, if either u or p1 (resp.

p2) is bijective the other is also.

11

 

5 e 6. Let q: S 4 fiR) denote the cannonical surjection
of (right) R-modules. This gives rise to a surjection

qO Id: SO54 OS. The kernel of qGId is the

f(R)
R-submodule of S 0 S generated byall elements of the form

f(r) a s = l a f(r)s, where r e R, s 6 S. It follows

P
that the sequence S 42 S ®R S EJEES fiR)

f(R) ®R S = 0 iff ker (q 011d) = S S S 1ff p2

is a surjection.

®R S 4 O is exact.

 

Therefore

 

S1m111arly S G f(R) - O 1ff p1 .is a surject1on.
We conclude this section with two examples of ring epi-
morphisms. More involved examples are constructed in Chap-

ter III.

Example 1 - Clearly any surjective ring homomorphism is an

epimorphism.

Example 2 - Let W be a multiplicatively closed subset of
a commutative ring R. Let ,i- be an arbitrary element of

RW. In RW ®R Rw we have the relations

€|H
the

r l_m -1: _1__ .1;

W

Thus the mapping r 4'? from R into RW is an epimor-

phism 'which is not surjective.

Remark - Example 2 can also be viewed as a special case of

condition 2 of theorem 1.1. Every element of Rw has a

decomposition of the form ,£-= 5-w'éy with both

€P1

WI}:
’1

and 'w l-= 1 in R-l.
w

12

4. Epimorphisms over Commutative Rings

 

This section presents some basic properties of epimor-
phisms. Most of these resultsare available in the litera-
ture but were derived independently by this writer. The
most important results of this section are propositions 1.4
and 1.6 giving conditions under which an epimorphism is
surjective. In chapter II proposition 1.6 is strengthened
by theorem 2.2.

Let f: R 4 S be a ring homomorphism. If f is an
epimorphism, S will be called an Epimorphic extension of
R. If R is commutative and f(R) is in the center of

S, f will be called a central epimorphism.

Proposition 1.2 - Let S be an epimorphic extension of a
commutative ring R and let A be any R-algebra. Then

S ®R A is an epimorphic extension of A.

Proof - We have the following sequence of A-module isomor-

phisms:

(SeRA)®A (S®RA)2 (se S)®

R R (A @% A) a S ® A.

R
Under this composition the monomial (s O a) S (s’ 8 a’)

is mapped to the monomial ss’ ® aa’ = (s G a)(s’ G a’).

Corollary 1.2.1 - Let u be an ideal of R and f: R 4 S

a central epimorphism. Then é%- is an epimorphic extension
JL
of m .

Proof — -jL 2 S ®

3 — ——
as R m under the map 3 8 r 4 sr .

l3

Corollary 1.2.2 - Let W be a multiplicatively closed sub-
set of R and S an epimorphic extension of R. Then SW

is an epimorphic extension of RW'
P f - S 3 S ® u d’ the ma ®'£ 4'§£
roo W" ‘RRW ner p s w w .

Of the many proofs of the next proposition, the short-

est one uses the concept of a faithfully flat module.

Definition - An R-module M is said to be faithfully flat
if it is flat and for any R-module N, the relation N @th= 0

implies N = 0.

Remark 1 — A direct sum of faithfully flat modules is faith-
fully flat and R is a faithfully flat R-module; therefone,

any free R-mdoule is faithfully flat.

Remark 2 - Let M be a faithfully flat R-module, and let
u denote the annihilator of M in R. Then m ®R M = O
which implies a = 0. Thus, a faithfully flat module is
faithful.
It should be mentioned that although a faithfully flat
module is both faithful and flat the converse is not true.
The module of rational numbers Q is both faithful and flat
as a Z-module but is not faithfully flat because 52? '82 Q = O.
The following lemma will be needed in chapter III [8,

chap. I, §3, n°l, proposition 1].

Lemma 1.3 - An R—module M is faithfully flat if and only

if for any pair of R-modules N and N3 the exactness of

14

the sequence 0 4 N’ E N is equivalent to the exactness of

the sequence 0 4 N’ ®R M €323 N ®R M.

We are now ready for the important

Proposition 1.4 - If R is a field and S is an_epimorphic

extension of R, then S e R.

Proof - Since S is an epimorphic extension of R, E%§7'® S

= 0. Every module over a field is free and is, therefore,

 

faithfully flat. Hence = 0. Since a faithfully flat

S
f(R)
module is faithful, S = f(R) a R.

There are also several proofs of the next proposition,
the first of which was given by Oort and Stooker [13, theor-
em 3.l]. The proof given here is relatively straightforward

and utilizes the following generalization of Nakayama's len-

ma [1, chap. IV, §7, lemma 2].

Lemma 1.5 - Let N be a finitely generated module over a
commutative ring R. An ideal u of R is such that
EN = N if and only if R = N + annihR (N) ‘where

annihRN = [r e R|rn = 0 Vn e N].

Corollary 1.5.1 - mN = N for every maximal ideal m of

R if and only if N = 0.

Corollary 1.5.2 - Let R be a commutative ring, S a fin-
itely generated R-algebra, and A a subalgebra such that
S = A + ms for every maximal ideal m of R. Then S = A.

We are finally ready to prove

15

Proposition 1.6 - Let R be a commutative ring and f: R 4 S
a central epimorphism. If S is a finitely generated R-

module, then f is a surjection.

Proof - Let m be a maximal ideal of R, then either

31-: O or 'fiL is an epimorphic extension of the field '3.

m8 m5 m
S R

Therefore, 8 = mS or EE-a-E. In either case S==R~1 + ms
for every maximal ideal m of R. Applying corollary 1.5.2
we have S = R-l = f(R).

The next three propositions are a somewhat heterogene-
ous collection. By using corollary 1.7.1, one can occas-
ionally consider only faithful epimorphic extensions. Prop-
osition 1.8, which considers epimorphic extensions of a do-

main, will be used in chapter III. Proposition 1.9 will not

be used elsewhere in the thesis but is of interest in itself.

Proposition 1.7 - Let f: R 4 S and g: S 4 T be ring ho-
momorphisms; R, S, and T need not be commutative.
1) If f and g are epimorphisms, so is g o f.

2) If g o f is an epimorphism, so is 9.

Proof - This follows directly from the definition of an epi-

morphism.
Corollary 1.7.1 - f: R 4 S is an epimorphism if and only
if S is an epimorphic extension of R-l.

Before considering epimorphic extensions of integral

domains we need some definitions. Let M be a module over

16

an integral domain R. An element m of M is called a
torsion element if there exists a non-zero element r of
R with rm = O. The set of torsion elements forms a sub-
module of M; this submodule is called the torsion submodule

and is denoted by M If, in addition, M is an R-algebra

T'
then MT is a two-sided ideal of M. If MT = 0, M is
said to be torsion-free over R.

Finally, an overring of an integral domain R is a

ring containing R and contained in the total field of quo-

tients of R.

Proposition 1.8 - Let S be a faithful, commutative, epi-
morphic extension of an integral domain R and let ST de-
note the torsion ideal of S. Then ST is a prime ideal

of S and 3' is an overring of R.
T

Proof - We first assume S is torsion-free over R. Let

KR and K denote the total quotient rings of R and S

S
respectively. KS is an epimorphic extension of S which
is an epimorphic extension of R. Thus KS is an epimor-

phic extension of R. If S is torsion-free over R ‘we

have R : KR g KS. Since KS is an epimorphic extension

of R, it is an epimorphic extension of the field KR.
Therefore, KS = KR and we have R c S c KS = KR. Hence

S is an overring of R.

Now let ST be the torsion ideal of 8. Since '2
T

is a faithful, commutative, torsion-free epimorphic exten-

l7

sion of R, it is an overring of R and as such is a do-

main. Therefore, ST is a prime ideal of S.

We conclude this chapter with an interesting proposition

relating epimorphisms to direct sums.

Proposition 1.9 - The external direct sum S e T of the
rings S and T is an epimorphic extension of R if and
only if S and T are epimorphic extensions of R and

S ®R T = T ®R S = 0.

Proof - If S and T are epimorphic extensions of R and

S ®R T = T ®R S = 0, then it follows from theorem 1.1

(4) that S e T is an epimorphic extension of R.
Conversely, suppose S 9 T is an epimorphic extension

of R. Let s 8 t e S 8R T. The kernel of

u: (some (S@T)4se'r

R

contains the element (3 + O) @ (O + t) which, therefore,

must be the zero element of (S e T) 8R

(S Q T) to (S @R S) e (S ®R T)

(S @ T). Under the
isomorphism from (S a T) ®R
e (T @R S)¢@ (T 8R T), (s + O) G (O + t) is mapped to

O + (8 ® t) + O + O ‘which is the zero element if and only

if s ®R t = O in S e T. Since 8 Q t ‘was arbitrary

R
we have, S @R T = O. Similiarly T ®R S = 0.
Therefore, S e T e.- (S e T) ®R (S e T) a- (S ®R S) @ (T @R T)

under the map (5 + t) 8 (s’+ t’) 4 83’ + tt’. It follows

easily that S and T are epimorphic extensions of R.

Chapter II

SUBALGEBRAS OF FINITELY GENERATED ALGEBRAS

Let R be commutative, A a faithful R-algebra which
is finitely generated as an R-module, and S a subalgebra

of A. This chapter contains three main results. The first

is that S is an epimorphic extension of R iff S R.
This is used to prove that if S is R-separable then it is
finitely generated as an R-module, thus answering the con-
jecture that motivated this thesis. Finally, a character-
ization of S is given if S is an inertial subalgebra

of A and A is commutative. A uniqueness statement is

a corollary of this characterization.

1. Central Epimorphisms

The results of this section rely on condition 2 of

theorem 1.1 which we restate in a more manageable form.

Theorem (1.1)’ - S is a faithful epimorphic extension of
R if and only if for every element 8 of S there exist
matrices o1 and 02 having entries from S and a mat-
rix 9 having entries from R such that (s) = 01 e 02
and the matrices ole and 902 have entries from R.
Here 9 is m x n, 01 is 1 x m, and 02 is n x 1.

Due to the unusual nature of the computations involved

in the next two results, some preliminary comments are in

order.

18

19

Let S be a ring and M a left S-module. Let 8
and ‘M denote the set of matrices having entries from S
and M respectively. Define addition in m and 3 and
multiplication in g as usual. If 0 = (sij) E 8 is
m x n and u = (mij) E m is n x p‘we define, quite natur-
ally, o H to be the m x p matrix whose i,j'-entry is
£21 sik mkj' If 0 is n x p and p is m x n ‘we de-
fine H - o to be the m x p matrix (otut)t, where
( )t denotes the transpose matrix. If x e M ‘we define
(o x) to be the matrix (sij x). Let c, o’ e s,
p, u’ E m, x, x’ E M, ‘Whenever these matrices have com-
patible dimensions, they satisfy the following properties:
1) on, u - o, and ox are all in. m
2) wW+oWn==ql+ofil 2W L;r(o+ofly=u-o+u-o

3) 0(u + u’) = Cu + Op' 3') (u + u’) -o

H
C
Q
+
C

4) (0 c’)u=0(o’ u)
5) (o o’)x = o(o’ x)

If S is commutative, then
6) u- (00’)=(u-0)-o’
7) o(o’x) = (ox) - 0’

Properties 2’, 3’ and 1-5 are immediate. Properties
6 and 7 follow from the equality (o o’)t = o’t at, which
holds whenever S is commutative.

We are now ready for the next two results and their

corollaries.

Proposition 2.1 - Let R be any ring, S a faithful epi-

morphic extension of R, and M a two-sided S-module. If

20

m E M is such that mr = rm for every r E R, then

ms = sm for every 5 e S.

Proof - Suppose m 6 M is such that mr = rm V r E R. Let
s 6 S and let 01, oz, 9 be the matrices of theorem (l.l)’.
Then (s)m = (C1 9 02)m = ol[(e oz)m] = ol[m(9 02)] =

01[(me)02] = 01[(em)02] = [(olemlo2 = [m(ole)]02 = m(s).

Therefore (s)m = m(s) ‘which implies sm = ms.

Corollary 2.1.1 - Let R be commutative, A a faithful
R-algebra, and S c A a faithful epimorphic extension of

R. Then S is contained in the center of A.

Proof - Since A is a faithful R-algebra, R is contained

in the center of A.

Corollary 2.1.2 - Let R be commutative and f: R 4 S a

central epimorphism. Then S is commutative.

Proof - S is a faithful epimorphic extension of f(R), and
f(R) is contained in the center of S.
Now we can prove an important generalization of propo-

sition 1.6.

Theorem 2.2 - Let R be a commutative ring, and let S be
a commutative, faithful epimorphic extension of R. If M
is a left S-module which is finitely generated as a left

R-module, then for any element m of M, Sm = Rm.

Proof - Let m be an arbitrary element of M, and let

(m, x1, x -, xn} be an R-spanning set for M. Then

2’

21

M = Rm + Rx + Rx + .-. + Rx
l 2 n

The following lemma and a straightforward induction on

n prove that Sm c Rm. Hence Sm = Rm.

Lemma 2.3 - Let R, S, and M be as in theorem 2.2. If
a left R-submodule N of M and an element x of M are
such that M = N + Rx, then N is a left S-submodule of

M.

Proof - Let s E S, n 6 N; it suffices to show sn 6 N.

There exist appropriately chosen matrices 01, 02 and e
such that 01 and 02 have entries from S, 019, 902, and
e have entries from R, and (s) = 01 6 02 . Since

M = N + Rx we can set ozn = “2 + ¢2x and olx = “1 + plx
where the “i are matrices with entries from N and the
$1 are matrices with entries from R.
Having established notation we proceed.
(8n) = 019(02n) = (019)u2 + 01(9 ¢ZX)
= (019m2 + (01x) - (a $2)
= (016m2 + “1 - (e ¢2) + (tlx) - (e ¢2)
= (Uleluz + u1 ' (9 $2) + $1 a (¢ZX)
= (ole)u2 + ul - (a $2) + $1 a (ozn) - ¢le “2
= (Ole)u2 + “1 - (e ¢2) + ¢1(e 02)n - ($19)u2

Since N is a left R-module the last equality expresses
(sn) as a sum of four l)<1 matrices each having entry from

N. Therefore sn e N. implying that N is a left S-module.

This completes the proof of theorem 2.2.

22

Corollary 2.2.1 - With R, S, M as in theorem 2.2, if there
exists an element m of M such that the mapping 5 4 sm

is injective, then S = R.

Corollary 2.2.2 - Let R be commutative, f: R 4 S a cen-
tral epimorphism, and A a ring containing S. If A is
finitely generated as a left R-module, under the structure

induced by f, then f is surjective.

Proof - S is a faithful epimorphic extension of f(R) and
is commutative by corollary 2.1.2. A is a finitely gener—
ated left f(R)-module and the mapping 3 4 s - 1 is in-

jective. Therefore S = f(R) by corollary 2.2.1.

2. Separable Algebras

This section contains a brief discussion of separable
algebras over commutative rings. The details of the mater-
ial mentioned here may be found in chapter II of [C].

Let R be a commutative ring, let A be an R-algebra,
and let A° denote the opposite algebra of A. The R-alge-
bra A ®R A° is called the enveloping algebra of A and
will be denoted by Ae. The ring A has a left Ae-module
structure defined by (Z ai 8 hi) - a = Z‘, ai a bi . The
surjection u: Ae 4 A defined by “(Z a:®bi) = Z aibi is

1 1
a left Ae-module homomorphism which is a ring homomorphism

if A is commutative. Let J = ker u and note that J

is the ideal of Ae generated by [a 8 1 - 1 8 a|a 6 A].

23

Proposition - The following conditions on an R-algebra A
are equivalent:
1) A is projective as a left Ae-module (under the above
structure).
2) O 4 J 4 Ae H A 4 0 splits as a sequence of left
Ae-modules. I
3) Ae contains an element e such that “(e) = l and
Je = 0.
An R-algebra satisfying the equivalent conditions of
the proposition is called a separable R-algebra.
NOtice that the element e described in condition 3
is an idempotent since e2 - e = (e-lGl) e E Je = O. The
element e of A ®R A° is called a separability idempotent
for A. Suppose el and e2 are separability idempotents
for A. Then “(e1 - e2) = “(e1) - g(ez) = l -1 = 0 so
that el - e2 6 J. Therefore 0 = (e1 - e2)el = e1 - eze1

implying that e e = e1: similiarly, ele2 = e Thus

2 1
el = e2 whenever A is commutative.

2.

The following properties of separable algebras will

be used in this chapter.

81) Let A be a separable R-algebra with center C. If
[0:J] denotes the right annihilator of J in Ae,
then u[0:J] = C.

82) If A is a separable R-algebra which is projective as

an R-module, then A is finitely generated as an R-

module.

24

83) If A is R-separable and S is any commutative R-
algebra, then A SR 8 is S-separable.

S4) Let A be an S-algebra and 8 an R-algebra with R
and S commutative.
i) If S is R-separable and A is S-separable, then

A is R-separable.

ii) If A is R-separable, then A is S-separable.

SS) If’ A is a separable R-algebra with center C, then
A is finitely generated and projective as a C-module
and separable as a C-algebra. In addition, C is
separable as an Realgebra.

$6) Projective Lifting Property - Let A be a separable
R-algebra and P an A-module which is projective as
an R-module (under the induced structure). Then P

is projective as an A-module.

3. Epimgrphisms gpg,Separable Algebras
Let R be commutative, and let A be an R-algebra.
If A is an epimorphic extension of R, then A is clear-
ly a separable R-algebra. This observation leads to some
interesting results.
1) Observe that 181 is a separability idempotent for A
iff ker u = 0 iff A is an epimorphic extension of R.
2) Let A be an epimorphic extension of R with center C.
Property 81 implies that C = u[0:J] = “A6 = A. This is
a second proof that an R-algebra whiCh is an epimorphic

extension of R must be commutative.

25

3) Suppose f: R 4 A is a central epimorphism and A is
projective as an R-module. Property 52 implies A is
a finitely generated R-module. Thus f is a surjection

by proposition 1.6.

We have seen that if R is commutative, f: R 4 S is
a central epimorphism, and S is either finitely generated
or projective as an R-module, then f is surjective. The
importance of the commutativity is shown by the following
construction discussed by Rdbson [17]. Let S be a ring
(with identity), u a proper right ideal of S and
R = [s E S|sfl C 3]. Then R is a subring of S. If Su = S,
then R # S and S is an epimorphic extension of R which
is both finitely generated and projective as a right R-
module.

Separable algebras have the projective lifting property.
The next proposition shows that algebras epimorphic over

their ground ring have an analogous flat lifting property.

Proposition 2.3 - Let S be an R-algebra which is an epi-
morphic extension of R. Let M be an S-module which is
flat as an R-module (under the induced structure). Then. M

is flat as an S-module.

Proof - Let 0 4 L 4 N be an exact sequence of S-modules.
In the notation of theorem 1.1 (10), 0 4 L’ 4 N’ is an ex-
act sequence of R-modules and M’ is a flat R-module. Hence
0 4 L’ e M' 4 N’®R m’ is exact. Therefore 0 4 L e M 4

R S

N @S M is exact by theorem 1.1 (10).

26

The next proposition gives a module-theoretic condition
under which a separable R-algebra is an epimorphic extension

of R.

Proposition 2.4 - Let R be a commutative ring and A an
R-algebra. Then A is an epimorphic extension of R if
and only if A is a commutative separable R-algebra and for

every right (resp. left) A-module M the map
( )1} M 4 HOmR(A,M)

defined by mL(a) = m - a is a right A-module isomorphism
(resp. ( )R: M 4 HomR (A,M) defined by mR(a) = a - m 15
a left A-module isomorphism).

Proof - Suppose A is a commutative separable R-algebra
and ( 55 M 4 HOmR(A,M) is a right A-module isomorphism
for every right A-module M. Then

x: HOmR(A,M) 4 HomR(A, HomR(A,M))

defined by

[001») mm» = Nab): 4) e HomRtA.M). a.b e A.

is an isomorphism. From [F, Chap. 8, theorem 4,page 165]
‘we have T: HOmR(A ®R A,M) 4 HOmR(A, HDmR(A,M)) defined by
[(T(¢))(a)](b) = 4) (a sh): t) E HomR(A 60R A,M). a.b e A.

is an isomorphism. Since A is commutative and R-separable,

the exact sequence 0 4 J 4 A ®R A H A 4 O splits as

A ®R A-modules. This gives rise to the split exact sequence

27

O 4 HomR(A,M) H’HomR(A ® A,M) 4 HOmR(J,M) 4 0.

where

[u’(¢)](a 8’ b) = «P o u) (a 0b) = 4> (ab): 4) e HomR(A,M), a,b€A.

This yields the commutative diagram

T
HomR (A 8 A,M) .€> H'omR (A, HomR (A,M))

 

I

H X

HomR (A,M)

where T and l are isomorphisms and u’ is injective.
Thus u' is also an isomorphism implying that HOmR(J,M) = 0
for every right A-module M. Picking M to be A @.A, we

see that J = 0 and so A is an epimorphic extension of

R.

The converse follows from corollary 2.1.2 and theorem
1.1 (9).

With the observations and easy results attended to, we
are finally going to solve the motivating problem of this

thesis.

Theorem 2.5 - Let R be a commutative ring and let A be
an R-algebra which is finitely generated as an R-module.
Then any separable subalgebra of A is also finitely gen-

erated as an R-module.

Proof - Let S be a separable subalgebra of A and let

C denote the center of 8. By property SS, S is finitely

28

generated as a C-module so it suffices to prove that C is
a finitely generated R-module. Again by property SS, C is

a separable R-algebra hence has a unique separability idem-

n .

potent e = Z’Xi 8 yi in C ®R C. Let R’ be the R-alge-
i=1

bra generated by {x1,--o,xn}. Then R’ is a subring of

C. Since A is finitely generated over R, each element
of A is integral over R: therefore, R’ is a finitely
generated R-module. (cf. chap. 3, sec. 1). We will use

corollary 2.2.2 to show that R’ = C. By property S4 ii),

C is a separable R’-algebra. It is easy to see that

n

e’ = 23x. ® y. in C ® , C is the unique separability
i=1 1 1 R

idempotent for C over R’. However, each xi is in R’

n

I _ _ __ ' I
so that e - 1 SEE; xiyi - 1 G u(e) — 1 8 1 1n C ®R C.
Thus C is an epimorphic extension of R’ with R’ c CcA
and A is finitely generated as a left R’-modu1e. Apply-
ing corollary 2.2.2 we have C = R’ which is finitely gen-

erated as an R-module. Hence S is finitely generated as

an R-module.

4. Inertial Subalgebras

Let R be a commutative ring and A a finitely gen-
erated R-algebra with Jacobson radical N. An inertial
subalgebra of A is an R-separable subalgebra S such that
A = S + N.

The next two lemmas are lemma 1.1 and remark 2.2 resp-

ectively of [D].

29

Lemma 2.6 - Let A be a finitely generated R-algebra with
radical N and let n(mA) denote the intersection of the

mA as m runs over all maximal ideals of R.

a) (rad R) - A s: N,

b) There exists a positive integer n such that an:n (mA).
c) If A is projective, (rad R) - A = 0 (mA).

d) If A is separable, N = n(mA).

Lemma 2.7 - Let A be a finitely generated R-algebra with
radical N. If S is a subalgebra of A and S + N = A

then rad S = S n N.

Lemma 2.8 - Let A be a finitely generated R-algebra with
radical N. If B and S are subalgebras with B c S, S

separable over R, and B + N = A, then B = S.

Proof - By theorem 2.5 S is a finitely generated R-algebra:
thus, by lemma 2.6 (d), rad S = n (m8) ‘where m runs over
all maximal ideals of R. Since A = B + N and B c S,
it follows that S is an inertial subalgebra of A. This
implies rad S = S n N. Therefore

S = A n S = (B + N) n S = B + (N n S) = B + rad S =

= B + m(m 5).

Hence S = B + ms for every maximal ideal m of R. Apply-

ing corollary 1.5.2 we have B = S.

Proposition 2.9 - Let A be a commutative, finitely generated
R-algebra with radical N. If S is an inertial subalgebra of

A, then S = n {BIB is a subalgebra of A and A = B + N].

30

Proof - Since S is a subalgebra such that A = S + N,
the intersection is contained in S.

Let B be a subalgebra of A such that A = B + N.
The subalgebra BS is a ring homomorphic image of B 8R 8
so is a separable B-algebra containing B. Therefore, ap-
plying lemma 2.8 with R = B, we have B = BS so that
S : BS = 8. Thus S c n[B|B is a subalgebra of A and

A = B + N].

Corollary 2.9.1 - Let A be a commutative finitely gener-
ated R-algebra. An inertial subalgebra of A, if it exists,

is unique.

Chapter III

INTEGRAL EPIMORPHIC EXTENSIONS

l. Integral Extensions

Let R be a subring of a commutative ring S. An
element 3 of S is said to be integral over R if there
is a monic polynomial f(x) in R[X] ‘with f(s) = O.

The following lemma is proven on pages 254 and 255 of

[I].

Lemma :3.l.- Let R be a subring of the commutative ring
S and let s be an element of S. The following statements
are equivalent:
1) s is integral over R.
2) The ring R[s] is a finitely generated R-module.
3) The ring R[s] is contained in a subring A of S
which is a finitely generated R-module.
4) S contains a finitely generated R-module M. with the
following two properties:
i) sM c M
ii) If y e R[s] is such that ym = O, vm e M, then

y = 0.

Remark - The definition of an integral element and the proof
of lemma 3.1 only require that R be commutative and S

be a faithful R-algebra. This observation will be used la-
ter on.

31

32

The elements of S which are integral over R form
a subring of S, containing R, called the integral Elg-
sure of R in S. If every element of S is integral over
R, S is called an integral extension of R. If the inte-
gral closure of R in 8 equals R, R is said to be
integrally closed in S. If R is integrally closed in its
total quotient ring, we simply say that R is integrally
closed.

Let R c S be commutative rings and assume S is an
integral extension of R. The following properties may be

found in [1, chapter V] .

11) For any ideal m of S, 3' is an integral extension
R
f --
° 9JfiR

12) For any multiplicatively closed subset W of R,

SW 2 S ®R RW 15 an 1ntegral extens1on of Rw.

I3) For every prime ideal P of R, there exists a prime
ideal 9 of S such that p = T n R. Moreover T

is maximal if and only if p is maximal.

2. Integrally Saturated Rings

Let R c S c A be rings with R and S commutative,
A an R-algebra, and S an epimorphic extension of R. Cor—
ollary 2.2.2 shows that if A is finitely generated as a
left R-module then S = R. This condition on A forces S
to satisfy certain internal conditions. Namely, lemma 3.1
and the remark following it show that S is an integral ex-

tension of R. Moreover, the proof of 4 e l in lemma 3.1

33

shows that if A is generated over R by n elements,
then every element of A, hence of S, satisfies a monic
polynomial in R[X] of degree n. It is natural to ask

if these conditions on S are enough to imply that S = R.
The examples of the next section show that fiiis is not true
in general: whereas, the results of this section show it

to be true if certain restrictions are placed on R.

The next lemma is fiéS, page 729 of [H].

Lemma 3.2 - Let A be a commutative R-algebra and S a

separable subalgebra. Suppose A contains an S-submodule
M such that A = S e M. Let e’ be a separability idem-
potent of 8 ® S and let e denote its image in A ®R A

R

under the inclusion map S 8R S 4 A ®R A. Then

s={xeA|(x®1—1ex)e=OinA®RA}.

Motivated by the terminology of Storrer [23], a com-
mutative ring R will be called ingggpglly sapuggped if the
only faithful.integral, epimorphic extension of R is R

itself.

Proposition 3.3 - An integrally closed domain is integrally
saturaped.

Proof - Let R be an integrally closed domain and let S be
a faithful integral epimorphic extension of R. Let ST de-
note the torsion ideal of S. By proposition 1.8 -§- is an

ST

overring of R 'which is an integral extension of R by

34

property 11. Since R is integrally closed this implies
R.= g- so that S = R.@ ST' Applying lemma 3.2 with

T
e’ = 1816 R®R R we have

R = [x e S|x ® 1 - 1 G x = 0 in S ®R S}.

Therefore, R = S since S is an epimorphic extension of
R.

The next task is to prove that a Noetherian ring is
integrally saturated. This result is not new. It is men-
tioned in the literature and follows easily from a theorem
due to Ferrand D2,§41 prop. 3]. However, neither Ferrand's
theorem nor this result are proven. The proof given here
first considers a complete local Noetherian domain then pro-
ceeds toward consideration of a Noetherian ring. This is a
simplified version of an unpublished proof given by E. C.
Ingraham.

We begin with a brief discussion of complete local rings;
a detailed discussion may be found in [E, chap. II, §l6, 17]
or [J, chap. VIII, §2,3]. Let N be an ideal of a commuta-
tive ring R. Define a topology on R by letting the sets
fl° = R and Mn, n > 0, be a neighborhood basis of zero:
that is, open sets are arbitrary unions of sets of the form
r + an where r e R. This is called the gfadio topglggy
on R and will be a Hausdorff topology if and only if

m n a n

“:0 m = o. If nn=o m

metric on R:

n = O, the following function is a

d(x,y) =-l- if x-y G an but x-y f ”n+1. n 2 0.

2n

35

Let R be a local ring with maximal ideal m such

n

that 0:: m = 0. If R is a complete metric space in the

0
metric induced by m, then R is called a complete 19931
ri g. If R is not complete it may be embedded in a com-

plete local ring by the following process. Let R be the

completion of the metric space R. Let f = lim {rm} and f‘
A 114:» 5
Q = lim [3n] be two elements of R where [rn] and [Sn]

n4m
are Cauchy sequences in R. It is easy to check that

[r + s ] and [r s ] are also Cauchy sequences in R.

n n

 

n n

. A . .

The operations r + Q = 11m [rn + SD] and 99 = 11m [rnsn}
n-Om n-oeo

are well-defined operations which endow R 'with a commuta-

“T

. . A . . . .
tive r1ng structure. R conta1ns a subring isomorphic to R
and

Q = mfi = [lim [mn]| {mu} is a Cauchy sequence in m }
n4a

is the unique maximal ideal of R. R is a complete local
ring called the completion of R and R is a flat R-module.

We will use the following properties of a Noetherian
local ring R.

l) The completion of R always exists since by Krull's

a n
n=0 m

2) The completion R of R isafaithfully flat R-module

Intersection Theorem n = O, [1, pg. 216].

since every R-module M is isomorphic to a submodule of
A

M ®R R, implying that M ®R R = 0 iff M = 0.

We will also need the following:

Definition - [E, pg. 30] - Let R and S be integral do—

mains with S an integral extension of R. We call S an

 

36

almost finite extension of R if the field of quotients of

S is finite dimensional over that of R.

Theorem [E, Thm. 32.1] - If R is a complete local Noether-
ian domain, then an arbitrary almost finite extension of

R is a finitely generated R-module.

 

Ft; '5:
Proposition 3.4 - Let R be a complete local Noetherian do- 5
main, Then R is integrally saturated. '
Proof - Let S be a faithful integral epimorphic extension
. . s J
of R and let ST be the torsion ideal of S. Then 'g; E;

is a faithful,integral, torsion-free, epimorphic extension

of R hence is an overring of R. Therefore, R and 'iL

ST
have the same field of quotients so that 'éi' is an almost
T
finite extension of R. From the above theorem ‘éi- is a
T
finitely generated R-module. Hence R = éi- by proposition
”T

l.6 so that S = R @ ST. Applying lemma 3.2 we have R = S.

Proposition 3.5 - A complete local Noetherian ring R is

integrally saturated.

Proof - Let p1, ..., pn be the minimal primes of R. Then
their product plpz-oopn is contained in the nil radical of
R [1, corollary to theorem 10, pg. 214]. Thus (plpzo-opn)k

= O for some integer k. Let A be a faithful integral

epimorphic extension of R. It follows from properties 13

A
piA
faithful integral epimorphic extension of the complete local

 

and 11 and corollary 1.2.1 that for each pi, is a

37

 

NOetherian domain Ji-. Therefore 18 - JL' so that

Pi P11" - 1’i
A = R + pi A for each pi. This implies that

k
A=R+ (plp2---pn)A=R

Proposition 3.6 - A Noetherian ring is integrally saturated.

Proof - Let S be a faithful integral epimorphic extension
of the Noetherian ring R. For every maximal ideal m of
R, let Rm denote the completion of the local Noetherian

ring Rm' Then Sm 8R Rm is a faithful integral epimor-
m

. . A . .
phic extens1on of Rm, hence equals Rm by propos1tion
3.5. Therefore, Sm = Rm for every maximal ideal m of

A

A
R, because S 8 R = R = R 8 fl and R is a faith-
m Rm m m m Rm m m

fully flat Rm-module. Hence S = R, for suppose s e S.
Then for each maximal ideal m of R there exists m E R- m
such that sm 6 R. Thus [r E Rlsr 6 R] is an ideal of R
contained in no maximal ideal, and so equals R. Therefore

3 = 3 ° 1 e R.

3. Examples of Integral Epimorphic Extensions

 

In each example the epimorphic property is shown by means
of the following argument. Let R and S be commutative
rings and f: R 4 S a ring homomorphism. Following the no-
tation of Storrer [23] we define the domipiog of R in s,
denoted Dom(R,S), to be the set of elements 3 of S such
that: If g and h are ring homomorphisms from S to a
ring T ‘with g o f = h o f, then g(s) = h(s). In other

words, the image of 3 under 9 is completely determined

38

by the action of g on f(R). It is easily shown that

Dom (R,S) is a subring of S containing f(R). Clearly

S is an epimorphic extension of R if and only if

S = Dom (R,S). Using the universal mapping property of

S (8 8, one can show that Dom (R,S) = [s E S|s®l=l®s in

R

S ®R S]. F-
In constructing the examples, we first construct a com-
mutative ring A and a subring R. Next, we let S be the

R-subalgebra of A generated by an appropriately chosen set

 

of elements of A. Finally, these generators are shown to
be in the subalgebra Dom (R,S). It follows that Dom(R,S)==S:
hence, S is an epimorphic extension of R. To show 8 E S
is in Dom (R,S) 'we show that there exist 51,32 6 S and

r e R such that s = 31 r s with slr and rs in R.

2 2
If such 81,82 and r exist, then in S 8R S we have
s 8 1 = slrs2 8 1 = 31 ®rs2 = s1r® 82 = l 8 slrsz=l®s.

Therefore 5 E Dom (R,S).

NOte - As in example 2 of chapter I, we are using a special
case of theorem 1.1 condition 2.

Both examples are subrings of discrete group rings over
a field where the group is an abelian, non-archimedean, or-
dered group. Before presenting the examples we investigate
properties of the group and of the field which will force
the rings to have desired properties.

Let G be any ordered abelian group and F any field.
Then the discrete group ring FG is an integral domain. As

is the case when G is the group of rational integers, it

39

will be convenient to denote an arbitrary element of FG by

X) f x0 where I is a finite subset of G and each f
o o

GGI

is in F.

H

If H is a subset of G ‘we denote by F the subset

of PG consisting of those elements Z) foxO 'where I is

GEI
a(finite) subset of H. If H is a submonoid of G, then

FH is a subring of FG containing the identity element of

PG.
Let 6+ be the monoid of nonnegative elements of G

and let H be a submonoid of 6+. The following properties

of H will force FH to have desirable properties.

1) There exists 9 6 H such that o e G and o 2 9 im-
plies o e H.

2) For every 0 6 G+ there exists a positive integer n
such that no 6 H.

3) There exists 8 6 H, B # 0, such that for every posi- '

tive integer n, '3 E H.

, +
4) For every 0 e G+ there ex1st o 6 G and n e H

 

1'02
such that o = 01 + n + 02 with 01 + n and n + 02 in
H.
H 6+
1) F and F hgye the same field of quotients
H 6+ (3"
Since F c F . it suffices to show that F is con-
tained in the field of quotients of FH. Let. Z) foxO be
+ OEI
an element of FG Then for each 0 e I, o 2 0 so that
o + 9 2 9. Hence, for every 0 e I, o + 9 e H, and
0+9
EfoxO = Z)f0(§F6-) is an element of the field of quotients
061 061 x

of PH.

40

6+ H
2) F is an integral extension of F

Let c E G+ and let n be a positive integer such that

n o E H. Then (x0)n = xno E FH so that x0 is a root of

+
Xn - xno 6 FH[X]. Now F6 is generated as a module over

F by [xolo e G+], and each x0 is integral over FH:

+
therefore, FG is an integral extension of PH.

3) FH is not a Noetherian domain

For every positive integer n, xfi/n e FH. This gives

rise to_an infinite ascending chain of principal ideals

(x5) : (XE/2) : (xB/B) S -°- 9 (xB/n) :

+
4) FG ig an epimorphic extension of FH

Let 0 6 6+ and let 01, 02 6 6+, n 6 H, be such that

 

'with 01 + n and n + 02 in H. .Then

0 = G + n-+ G
10 2 o + o o
o o
x0 = x 1 x“ x 2 with x 1, x 2 6 FG and xn,x 1x“, x‘nx2
4..
all in FH. Thus Dom (FH,FG ) contains [xolo E G+]' and
+ +
so FG = Dom (FH,FG ).

We now give an example of an ordered abelian group G
and a submonoid H of G+ satisfying properties 1 througr

4. Then for any field F, FH ‘will be a non-Noetherian do-

+
main having FG as a faithful integral epimorphic exten—

+
sion and FH # FG

Example 1 - Let G be the abelian group Q a Z with lexi-
cographical ordering, i.e. (p,m) < (q,n) iff p < q or
p=q and m<n. Let H: [(q,n)|q€[0,l]and n20] U
[(q,n)lq > 1 and n is arbitrary]. Clearly H is a proper

submonoid of 6+. The shaded area of figure 1 represents H.

 

 

Z.
i / .x’
.F.//' /// ’//
,"'.’// /”
.4 . .,
/" -"'/.///1//’/ /
/. __. _/’.-”'/ //
, ./ /' /
y’ / x/ // ,
/ / / I .— 1’
// /{/ /'/,r ,/ /.I/
// -"’ / "I, I./ i
/ / / ."c /'/ ’/
,// z/ // ’ _ '/ . "It ’
///L/ J‘/ / ,7" /"” ,.//
(100) :i // //// W, I”
’y/ /’ ,-/ '/-’ ..’
I ’,./ ,"' / .. /
'/ /‘/ l/‘f //
'J"///I ,"f///'/' H,-
,/ ”
s/ ,/'" ,/ /,,/ ,.»-/
///'/ ’ ’ 'Il’ ’/" 7’
LI/ I ”"/” '/,/'//:’
I r/ I
‘I’:/,/ /"///r‘.” / /////
I, /’ ,/“
i -»x/‘/,///
-. '// ,
I/
Figure l.

1) Let 9 = (2,0). If 0 = (q,n) e G+' with o 2 9, then
q 2 2 so that o e H.

2) Let c = (q,n) E G+ - H, then q 6 (0.1] and n < 0.
Let m be a positive integer such that mq > 1.
Then mo = (mq,mn) 6 H since mq > 1.

3) Let B = (1,0), then for every positive integer n
%= ($7.0) e a.

4) Let G = (gm) 6 6+ - H. Set 01 = 02 = (aim) and
n = (0,-n). Then 0 = 01 + n + 02, 01,02 6 6+, 01 + n =

n + o = (g, 0) 6 H and since n < 0, n = (0,-n) e H.

2
For the second example we exhibit an abelian ordered

group G, a submonoid H of 6+, an element n of H,

and a subset A = [aijli = 0,1,2,---j = l,2,3,---,Zi] of 6+

with the following properties:

42

l) V a.. 6 A, oi

1] + n E H.

j

2) v aij e A, aij g H but Zaij e H.
3) V aij E A, aij = ai+1,j + n + ai+1,j+21 (w1th ai+1,j in
and n + ai+l,j+21 1n H by l).

+
Let S be the FH-subalgebra of PG generated by

 

a..
[x 13Iaij 6 A}. As in the first example, consequences of the ‘
above properties are:
l) FH and '8 have the same field of quotients.
aij 2 201' H

2) For each aij e A, (x ) = x 3 e F so that S is

an integral extension of FH. E
3) S is an epimorphic extension of PH.
Example 2 - Let G1 = Z a Z 6 . - - with lexicographical order

and let G = G 9 Z again with lexicographical order. Let

1
H be the submonoid of 6+ generated by all elements of the

form (g,n) where g 6 G1 has only nonnegative entries,
n e Z is nonnegative if every entry of g is 0 or 1
and n is arbitrary if at least one entry of g is neither

0 nor 1.

For i = 0,1,2,... and j = l,2,3,...,21, define ‘ij
to be the element of G1 ‘whose k'th entry is 1 if k s j

mod (21) and 0 otherwise.

43

5‘01 = (l,l,1,... )
a11 = (l,0,l,0,l,0,... )
a12 = (O,l,O,1,0,l,... )
321 = (1,0,0,0,1,0,0,0,..)
a22 = (O, l,0,0,0,1,0,0,..)
a23 = (0,0,1,0,0,0,1,0,..)
a24 = (0,0,0,1,0,0,0,1,..)
etc.

Figure 2.

Notice that for i = 0,1,2,... and j = l,2,3,..,2
aij = ai+1,j + ai+l,j+21'
Now, for i = 0,1,2,... and j = l,2,3,...,2 set

— — +- =
aij — (aij. 1) E G H. n (0.1) E H. and

. . i
A _ [aijll - 0,1,2,... 3 _ l,2,3,...,2 }

Then for every oi. e A:

J

1) aij + n = (aij.0) e H

2) aij g H but 2 aij = (zeij,-2) e H.

3) aij = (aij,-l) = (ai+l,j’-1) + (0,1) + (ai+l,j+21’-1)
= ai+l,j + n + ai+1,j+2i°

+
Let S be the FH-subalgebra of PG generated by

a..
[x 13| aij E A], then S is an integral epimorphic exten-
sion of FH. Let K be the submonoid of G+ generated by

H U A. It is easily seen that S = FK and for each B e K,
28 e H. By picking F to be a field of characteristic 2,

we have that for every element 5 = ZEfaxfi of S = FK,

44

2

s = ( fo x5)2 = Z)f2x2B is in FH. Therefore, S is an

5 B

integral epimorphic extension of FH and every element of

S satisfies a monic polynomial in FH[X] of degree 2.

Remark 1 - Much of the research concerning epimorphisms of
rings has been restricted to flat epimorphic extensions of
commutative rings. The examples of this section are commu-
tative epimorphic extensions which are not flat. This is
because an overring S of an integral domain R which is
an integral extension of R is flat iff S = R [G, prop. 2,

pg. 797].

Remark 2 - Example 1 comes close to answering an open ques-
tion. Let R be an integral domain with field of quotients
K and let R. denote the integral closure of R in K.

If R #‘R, it is possible for R. to be a separable R-
algebra? It is the opinion of this writer that a construc-
tion similar to example 1 can be made in which R. is an

epimorphic extension of R.

BIBLIOGRAPHY

 

B IB LIOGRAPHY

Ring Theory

G. Azumaya, "0n maximally central algebras", Nagoya
Math J..; (1951), 119-150.

N. Bourbaki, Algebra commutative, chapters I-II,
actualites Sci. Ind. no. 1290, Hermann, Paris, 1962.

F. DeMeyer and E. C. Ingraham, Separable algebras over
commutative rings, Springer-Verlag, New York, 1971.

E. C. Ingraham, "Inertial subalgebras of algebras over
.commutative rings", Trans. Amer. Math. Soc. 124 (1966),

M. Nagata, Local Rings, Interscience, New York, 1962.

D. G. Northcott, Homological Algebra, Cambridge Univer-
sity Press, Cambridge, 1960.

F. Richman, "Generalized quotient rings", Proc. Amer.
Math. Soc. 12 (1965), 794-799.

0. E. Villamayor and D. Zelinsky, "Galois theory for
rings with finitely many idempotents", Nagoya Math J.
.21 (1966), 721-731.

0. Zariski and P. Samuel, Commutative Algebra,‘Vol.
1, Van Nostrand, Princeton, N.J., 1958.

. Commutative Algebra, Vol. 11, Van
Nostrand, Princeton, N.J., 1960.

 

Epimorphisms

D. Ferrand, "Epimorphismes d'anneaux e source noetherienne
et monomorphismes de schemes", C. R. Acaqg Sci. Parig,
set. A-B, 266 (1968), A319-A321. MR 39 #5197e.

, "Epimorphismes d'anneaux et algebras separ-
ables“, C. R. Acad, Sci. Parig, 5gp, A-B. 265 (1967),
A411-A414. MR 39 #5628.

45

10.

11.

12.

13.

14.

15.

16.

46

G. D. Findlay, "Flat epimorphic extensions", Math. Zeit.
118 (1970), 281-288.

M. Hacque, "Remarques sur 1es epimorphismes plats
d'anneaux", C, R. Aged. Sci, Paris 86%. A-B, 268 (1969),
A1447-Al450.

 

. "Sur l'existence de l'epimorphisme injectif
plat s gauche maximum", C. R. Acad. Sci. Paris, set A-B,
271 (1970), A1218-A1221. MR 43 #4289.

J. M. HOwie and J. R. Isbell, "Epimorphisms and Domin-
ions 11", J. Algebra Q (1967), 7-21.

J. R. Isbell, "Epimorphisms and Dominions IV", J. London
Math. Soc., (2), 1 (1969), 265-273. MR 41 fié1774.

J. T. Knight, "0n epimorphisms of non-commutative rings",

Proc. Cambrid e Philos. Soc. 99 (1970), 589-600.
MR 42 3 4595.

D. Lazard, "Epimorphismes plats d'anneaux", C. R. Acad.
Sci. Paris. Ser A-B 266 (1968). A3l4-A316. MR 39 #197c.

P. Mazet, "Generateurs, relations et epimorphismes

d'anneaux", Q. R. Acad. Sci. Paris, Se: A-B 266 (1968),
A309-A311. MR 39 # 197a.

, "Characterisation des epimorphismes par rela-
tions et geherateurs", Seminaire d'al ebre commutative:
1967168. Les epimorphismes d'anneaux, (2), SecrEEariat
mathematique, Paris, 1968.

J. Oliver, "Anneaux absolument plats universels et
epimorphismes d'anneaux", C. R. Acad. Sci. Paris, Ser
A-B 266 (1968), A3l7-A318. MR 39 fiél97d.

F. Oort and J. R. Strooker, "The category of finite bi-
algebras over a field? Proc. Kgn. Ned. Ak. v. ng, A70
(1967), 163-169.

N. Popescu and T. Spircu, "Quelques observations sur 1es
epimorphismes plats (e gauche) d'anneaux", J. Algebra
lg (1970), 40-59. MR 42 #323.

. "Sur les epimorphismes plats d'anneaux",
C. R. Acad. Sci. Paris Ser A-B 268 (1969), A376—A379.
MR 39 #4 4227.

 

J. Raynaud, "Epimorphismes plats d'anneaux et proprietes
de transfert", C. R. Acad. Sci Paris ser A-B 270 (1970),
Al7l4-Al717. MR 42 #[1860.

 

 

17.

18.

19.

20.

21.

22.

23.

24.

25.

47

J. C. Robson, "Idealizer Rings", Proceedings of a confer-
ence on Rinquheory held in Park City, Utah, Academic
Press, New York, 1972.

N. Roby, "Sur les epimorphismes de la categorie des
anneaux", C. R. Acad. Sci. Paris, SGT A-B 266 (1968),
A312-A313. MR 39 # 197b.

"Diverses caracteTisations des epimorphismes",
Seminair d'alqebre commutative: 1967/68. Les epimor-
phismes d'anneaux. (3), Secretariat Mathématique, Paris,
1968.

P. Samuel, Seminaire d'alqebre commutative. diriqé par
Piegpe Samuel: 1967/68. Les epimorphismes d'anneaux
Secretariat mathématique, Paris, 1968. MR 39 #56867.

L. Silver, "Noncommutative localization and applications",
J. Algebra 1 (1967), 44-76.

H. H. Storrer, "A characterization of Prufer domains",
Canad. Bull. Math. 12 (1969), 809-812. MR 40 #47238.

, "Epimorphismen von kommutativen Ringen“,
Comm. Math. Helv. 43 (1968), 378-401. MR 39 #44137.

"Epimorphismes d'anneaux et anneaux des

fractions", C. R, Acad, Sci, Paris, Ser A-B 266 (1968),
A322-A323.

 

"Sur 1es epimorphismes dans la categorie
des anneaux commutatifs", C. R. Acad. Sci. ParisL SQ;
A-B 266 (1968), A263-A265.