INERTIAL SUBALGEBRAS OF ALGEBRAS
SEPARABLE OVER THEIR CENTERS

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
NICHOLAS STEVEN FORD
1972

 

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thesis entitled
INERTIAL SUBALGEBRAS 0F ALGEBRAS
SEPARABLE OVER THEIR CENTERS

presented by

NICHOLAS STEVEN FORD

has been accepted towards fulfillment
of the requirements for

PH.D. MATHEMATICS

degree in

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ABSTRACT

INERTIAL SUBALGEBRAS OF ALGEBRAS
SEPARABLE OVER THEIR CENTERS

BY

Nicholas Steven Ford

Let A be a finitely generated algebra over a commuta-

tive ring R. A separable R-subalgebra r of A having

the property that T + N = A, where N is the Jacobson

radical of A, is called an inertial R-subalgebra of A

E. C. Ingraham, Trans. Amer. Math. Soc. 124 (1966)].
Theorem: Suppose A is a finitely generated R-algebra

which is separable over its center C. If A possesses an

inertial R-subalgebra F, then

(1) S = F n C is an inertial R-subalgebra of C-

(2) A m T ® C.
S

The inertia subgroup, denoted IA' of a group G of
R-automorphisms of an R-algebra A is defined to be

[c E G|o(a) - a E N = rad A V a E A}.

Theorem: Suppose A is a finitely generated, faithful

R-algebra which is separable over its center C. Assume C

has no idempotents other than 0 and l, and that A posses-

ses a finite group G 'of Reautcmorphisms which restricts faith-

Nicholas Steven Ford

fully to C in such a way that the G-fixed subring of C

is R. If I = I and II is a unit in R, then the

A c Al
existence of an inertial R-subalgebra of C implies the
existence of an inertial R-subalgebra of A.

The uniqueness statement is said to hold for an R-alge-
bra A provided any two inertial R-subalgebras B and B*
of A are isomorphic via an inner automorphism of A gen-

erated by l — n, where n E N = rad A. We denote the

Brauer group of a commutative ring R by 8(R).

Theorem: Suppose R is a semi-local ring and C is
a finitely generated, commutative R-algebra with inertial
R-subalgebra S. Then the natural mapping 8(5) 4 $(C) is
a monomorphism if and only if the uniqueness statement holds
for every R-algebra which is central separable over C.

The generalized quaternion algebra, denoted (C,x,y),
over a commutative ring C is a free C-module having basis
{1 E C,a,B,aB} with multiplication induced by a2 = x E C,
32 =yEC, and Ba = -aB. We denote the n-by-n matrices over
a commutative ring R by 'mn(R), and the localization of
the integers a at the maximal ideal (p) by z .

P

2

Example: Let R = 25 and C :55 a 52 i, where i =

5
-1. Then (C,-l,-l) is an R-algebra which possesses two
non-isomorphic inertial R-subalgebras, namely: *m2(R) and

(Ro-l.-1).

INERTIAL SUBALGEBRAS OF ALGEBRAS
SEPARABLE OVER THEIR CENTERS

BY

Nicholas Steven Ford

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Mathematics

1972

To My Parents.

ii

ACKNOWLEDGEMENTS

I wish to thank Divine Providence for Being a Source
of Strength and Inspiration in the dark and lonely hours
when this dissertation was born. I am grateful to Profes—
sor E. C. Ingraham for directing these investigations,
and for his invaluable assistance and timely suggestions,
especially with regard to example 3.9. I thank him also
for his kind and patient guidance not only during the re-
search, but also throughout my entire career as a graduate
student. I wish to offer special thanks to Dr. W. C. Brown
for numerous helpful conversations. I am also deeply appre-
ciative of the encouragement and moral support given me by
my fiancee Ann Bilsborrow and my parents Michael and Rosalie
Ford. Finally, I would like to express my gratitude to
Michigan State University and the National Science Founda-

tion for their financial support over the past six years.

iii

TABLE OF CONTENTS

Page
Introduction . . . . . . . . . . . . . . . . . . . . 1
Chapter
I. PRELIMINARIES . . . . . . . . . . . . . . . 4
§1. Basic Ring Theoretic Results . . . .
§2. Separable R-Algebras and Related
Properties . . . . . . . . . . . . .
§3. Background and Properties of Inertial
Subalgebras . . . . . . . . . . . . .
II. EXISTENCE OF INERTIAL SUBALGEBRAS OF ALGE-
BRAS SEPARABLE OVER THEIR CENTERS . . . . . 16
III. UNIQUENESS OF INERTIAL SUBAIGEBRAS OF
ALGEBRAS SEPARABLE OVER THEIR CENTERS . . . 39
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . 53

iv

INTRODUCT ION

Let A be a finitely generated algebra over a commu-
tative ring R. A separable R-subalgebra B of A having
the property that B+N = A, ‘where N is the JaCObson rad-
ical of A, is said to be an inertial R-subalgebra of A.

The present work deals with inertial subalgebras of
those finitely generated algebras which are separable over
their centers. We investigate the relationship between in-
ertial subalgebras of these algebras and inertial subalge-
bras of their centers. We also exhibit conditions under
which the question of isomorphism of inertial subalgebras
of such algebras can be formulated as a question involving
Brauer groups (Definition 3.1).

Chapter I consists of definitions and basic results
which are needed in the body of the thesis.

In Chapter II we show that the existence of an inertial
R-subalgebra of a finitely generated R-algebra which is separ-
able over its center always implies the existence of an iner-
tial R-subalgebra of the center. The main result of this
chapter is a partial converse. Suppose A is a faithful R-
algebra possessing a finite group G of R-automorphisms. The
inertia subgroup IA of G is defined to be

[0 E G|o(a)-a e N = rad A V a 6 A].

l

Theorem: Let A be a finitely generated R-algebra
which is separable over its center C. Suppose C has no
idempotents except 0 and l, and that A possesses a fi-
nite group G of R-automorphisms which restricts faithfully
to C in such a way that the G-fixed subring of C is R.
If IA = IC and IIAI is a unit in R, then the existence
of an inertial R-subalgebra S of C implies the existence
of an inertial R-subalgebra r of A. Furthermore, A=~r ® C.

One notes that this result simultaneously extends a tgeo-
rem of Ingraham ([12], theorem 2.10) and a theorem of DeMeyer
([7], theorem 1.5).

In ChapterIII‘we investigate the uniqueness (up to iso-
morphism) of inertial R-subalgebras of finitely generated R-
algebras which are separable over their centers. We say the
uniqueness statement holds for a finitely generated R-alge—
bra A provided any two inertial R-subalgebras B and B*
of A are isomorphic via an inner automorphism of A gen-
erated by l - n ‘where n 6 N = rad A. We denote the Brauer
group (definition 3.1) of a commutative ring R by 8(R).

Theorem: Suppose R is a semi-local ring and C is
a finitely generated, commutative R-algebra with inertial
R-subalgebra S. Then the natural mapping 3(8) 4 3(C) is
a monomorphism if and only if the uniqueness statement holds
for every R-algebra which is central separable over C.

Chapter III is concluded with the only known example of

an R-algebra which possesses non-isomorphic inertial R-sub-

algebras. The example will be seen to be a finitely gener-
ated R-algebra which is separable over its center. It is
also endowed with a finite group of automorphisms which sat-

isfies all the conditions mentioned in the first theorem of

this introduction.

Chapter I

PRELIMINARIES

In this chapter we introduce notation, establish conven-
tions, and present the basic results needed in the remainder
of the thesis. We supply proofs only for those results whose
proofs are not clear in the literature. References are giv-
en for all other results. For clarity we divide the chapter
into three sections: §l - Basic Ring Theoretic Results, §2 -
Separable Algebras and Related Properties, §3 - Background
and Properties of Inertial Subalgebras.

All rings are assumed to be associative and to possess
an identity element 1. A commutative ring is said to be
lgggl_if it possesses a unique maximal ideal, and semi:lgg§l
if it possesses only finitely many maximal ideals. It will
be our convention that ring homomorphisms map the identity
element to the identity element. By an algebra A over a
commutative ring R we mean a ring A together with a ring
homomorphism e from R into the center of A. Many of
the algebras we have occasion to deal with are such that e
is, in fact, a monomorphism. Such algebras are said to be
R-faithful. When we say an algebra is finitely generated or
Draientixe, we mean the algebra is finitely generated or pro-

jective as a module over its ground ring R.

§l Basic Ring Theoretic Results

Let A be a ring or an R-algebra. We will denote the
Jacobson radical of A by rad A and the center of A by
Z(A). When there is little chance for confusion, we will use
the abbreviated notation N = rad A, C = Z(A), and n =
rad C.

Lemma 1.1: (Original Nakayama Lemma) Suppose A is a
ring and M is a finitely generated A-module. Then
(rad A)-M = M if and only if M = (0).

Lemma 1.2: (Generalized Nakayama Lemma [20]) If R is
a commutative ring and M is a finitely generated R-module,
then an ideal m of R has the property that m°M = 14 if
and only if annihR(M) + u = R, ‘where annihR(M) = (reler-M
= (0)].

Lemma 1.3: (Ingraham [12]) Let A be a finitely gen-
erated R-algebra and let n(mA) denote the intersection of
mA as m runs over all maximal ideals of R.

(a) (rad R)-A : N.

(b) There exists a positive integer n such that

N“: MM).
(c) If A is R-projective, then (rad R)-A = n(mA).
(d) If A is R-separable*, then N = 0(mA).

(*) See definition 1.11

Proposition 1.4: An R-subalgebra B of a finitely generated

R-algebra A has the property that rad B = B n N.

Iggggf: It follows from the corollary on page 126 of
[2] that B n N : rad B. Suppose (rad B)-A ¢ N. Then there
exists some maximal right ideal 7/1 c A such that (rad B)-A+771 =
A. Hence (rad B)-A/m = Aflm , and therefore Aflm = (O) by
lemma 1.1. This is a contradiction, and so rad B : (rad B)-A
C N.

Let R be a commutative ring, and let S c R be a mul—
tiplicative system. The module of quotients ([8], page 25)
of an R-module M ‘with respect to the multiplicative system

S is generally denoted M However, when S is the com-

S‘
plement of a maximal ideal m of R, it is customary to
denote the module of quotients of an R-module M ‘with respect
to the multiplicative system R - m by Mm' The module of
quotients Mm is naturally isomorphic to M g Rm as Rm-
modules where Rm is the ring of quotients with respect to
the multiplicative system R - m. The ring of quotients Rm
is a local ring with unique maximal ideal m-Rm such that
Rm/m'Rm 3 R/m, and'we refer to Rm as the localization 9;

the ring R at the maximal ideal m. When viewed as an R-

module, the module of quotients Rm is R-flat.

Proposition 1.5: If A is a finitely generated R-algebra,

then Nm ; rad (Am) for every maximal ideal m of R.

Proof: Since Rm is R-flat, it is clear that Nm : Am'
One can show that the maximal right ideals of Am have the
form. mm for certain of the maximal right ideals ‘m of A.

Since each mm D Nm' one sees that Nm s: n mm = rad (Am) .

It is well-known (see, e.g., proposition 1.4.1 of [8])
that a module over a local ring is finitely generated and
projective if and only if it is free of finite rank. Thus if
M is a finitely generated and projective R-module, then Mm
is free of finite rank over Rm' This rank is referred to
as the m-rank of M and is denoted rankm(M). If there ex—
ists an integer n such-that rankm(M) = n for all maximal
ideals m of R, we say the rank of M (denoted rankR(M))

is well-defined and equal to n.

Definition 1.6: A commutative ring R is said to be

QQBDQQEQQ if it possesses no idempotents except 0 and 1.

Theorem 1.7: ([8], page 32) If R is a (commutative)
connected ring, then for every finitely generated and pro-
jective R-module M the rank of M is well-defined

and finite.

Proposition 1.8: ([4], page 141) A finitely generated and
projective module of well-defined rank over a semi—local

ring R is free (of finite rank).

Corollary 1.9: A finitely generated and projective mod-

ule over a connected, semi-local ring is free of finite rank.

§2 Separable R-algebras and Related Properties

The concept of a separable R-algebra as it is presented
here was first introduced by M. Auslander and 0. Goldman in
[1] which was published in 1960. Using this concept,the
authors were able to extend the definition of the Brauer
group of a field to that of any commutative ring.

Let A be an R-algebra, and denote its opposite alge-
bra by A°. We refer to Ae = A 3 A° as the enveloping alg-
ebra of A. The algebra A has Ehe structure of a left Ae-
module induced via (a O a’)-b = aba’. There is a natural
Ae-module map I4 from Ae to A induced by u(a®a’) =aa’ .

Let us denote the kernel of u by 3. we then have the fol-

lowing exact sequence of Ae-modules:
O434AeI-1IA4O

Propgsition 1.10: The following conditions on an R-algebra
A are equivalent:
(a) A is projective as a left Ae-module under the
above structure.
(b) O 4 g 4 Ae g A 4 0 splits as a sequence of left
' Ae-modules.

(c) Ae contains an element 6 (necessarily an idempo-

tent) such that 36 = 0 and u(€) = 1.

Definition 1.11: An R-algebra A is called R-separable

if it satisfies the equivalent conditions of prop. 1.10.

We now list those formal properties of separable R-alge-
bras which we will have occasion to employ in the succeeding
chapters. These results can also be found in Chapter II of

the notes of DeMeyer and Ingraham [8].

Proposition 1.12: (Transitivity) Let S be a commutative,
separable R-algebra and let A be a separable s-algebra.
Then A is naturally an R-algebra and is R-separable. If,
on the other hand, A is given to be a separable R-algebra
and S is any R-subalgebra of the center of A, then A is

separable over S.

Proposition 1.13: An R-algebra A is separable if and only
if A is separable as an algebra over its center Z(A) and

Z(A) is separable as an R-algebra.

Proposition 1.1 : Let A be a separable R-algebra and let

 

U be a two-sided ideal of A. Then A/u is a separable R-

algebra. Furthermore, Z(A/U) =[Z(A) + uJ/u.

Proppsition 1.15: If A is a separable R-algebra and C is
a commutative R-algebra, then A g C is a separable C -alg-
ebra with center Z(A O C) m Z(A) O C. Conversely if C is
a commutative R-algebr: which possZsses R as a direct sum-
mand and A is an R-algebra such that A a C is C-separable,

R
then A is R-separable.

Proposition 1,16: If A is a separable R-algebra, then ev-
ery A-module which is R-projective is A—projective.~
A faithful R-algebraA is said to be central separable

over R provided it is R-separable and Z(A) = R. In view

10

of proposition 1.4, every separable R-algebra is central
separable when considered as an algebra over its center.
Central separable R-algebras are generalizations of central
simple algebras over a field, and many of the structure
theorems for central simple algebras have analogues in the

central separable setting.

Proposition 1.17: If A is central separable over R, then

A is R-projective.

Proposition 1.18: If A is central separable over R, then

R is a direct summand of any R-subalgebra B of A.

‘gpopositionql,l9: Suppose A is a central separable R-
algebra. Then there is a one-one correspondence between
ideals a of R and two-sided ideals u of A given

by a 4 QA and N 4 u n R.

Proposition 1.20: If A is a central separable R-algebra,

then N = nA where N = rad A and n = rad R.

We conclude this section with two results on separabil-
ity which are of a more advanced nature than those preceding.

The following theorem of S. Endo and Y. Watanabe pro-
vides an immensely useful characterization of separability

over a commutative ring.

Theorem 1.21: (Endo‘Watanabe [10]) The following state-
ments concerning a finitely generated R-algebra are equiva-
lent:

(a) A is a separable R-algebra .

11

(b) Am is a separable Rm-algebra for every maximal
ideal m of R.
(c) A/mA is a separable R/m-algebra for every maxi-

mal ideal m of R.

The final result of this section has been proved only
recently by Dean Sanders, a fellow graduate student at Michi-

gan State.

Theorem 1.22: (Sanders [16]) A separable R-subalgebra
of a finitely generated R-algebra is itself finitely gen-

erated.

12

§3. Background and Properties of Inertial Subalgebras

The following theorem due to J. H. Maclagan Wedderburn
appeared in a paper titled "0n Hypercomplex Numbers" pub-
lished in the London Mathematical Society Proceedings, Ser
2, vol. 6, Feb. 1908 - Jan. 1909.

The Oriqina1_Weggerburn Principal Theorem: If A is
an algebra in which every element, which has no inverse, is
nilpotent, it can be expressed in the form A = B + N ‘where
B is a primitive algebra and N is the maximal nilpotent

invariant subalgebra.

Remark: In Wedderburn's terminology, an "algebra" is
a finite dimensional algebra over a field, a "primitive alg-
ebra" is a division ring, and an "invariant subalgebra" is
an ideal (so that N mentioned above is therflJ.radica1 =
Jacobson radical of A).

The theorem was subsequently refined and generalized by
Wedderburn and others, most notably A. Malcev who contributed
a "uniqueness statement". Using Hochschild cohomology, C.

W. Curtis and I. Reiner present in [5] a unified version

which is usually referred to as:

The Wedderburn-Malcev Theorem: Let A be a finite
dimensional algebra over a field F such that A/N is F-
separable. Then A possesses a separable F-subalgebra B
such that A = 8 ® N. Furthermore B is unique up to an
inner automorphism of A generated by an element of the

form 1 - n for some n in N = rad A.

13

The following definitions appear in a paper titled
"Inertial Subalgebras of Algebras over a Commutative Ring"

by E. C. Ingraham [12].

Definition 1.23: Let A be a finitely generated alge—

 

bra over a commutative ring R. A separable R-subalgebra
B of A such that B + N = A (where N = rad A) is said

to be an inertigerégubalgebra of A.

‘nginition 1.24: A commutative ring R is said to be
an inertial_coefficient ping (I.C.-ring) if every finitely
generated R-algebra A such that A/N is R-separable pos-

sesses an inertial R-subalgebra.

Definition 1L;§: The uniqueness statement is said to
hold for a commutative ring R if for any two inertial R-
subalgebras B and B’ of a finitely generated R-algebra

-1
A there exists an element n E N such that (l-n)B(l-n) ==Bf

Remark: In view of Theorem 1.22, we see that every R-
inertial subalgebra of a finitely generated R-algebra is
itself finitely generated.

In the context of these definitions,Azumaya's main theo-

rem [2] can be restated as:

Theorem: (Azumaya) A Hensel ring is an inertial coef-

ficient ring for which the uniqueness statement holds.

We find the following property of inertial subalgebras
quite useful.
Proposition 1.26: Suppose A is a finitely generated R-

algebra with inertialrvsubalgebra B. Then there exists a

l4

one-one correlpondence between the maximal left (resp. right)
ideals of A and the maximal left (resp. right) ideals of

8 given by m 4‘m n B and m 4 m + N where N = rad A.

2329;: Since N’= n m ‘where m runs over all the max-
imal left (resp. right) ideals of A, there exists a one-one
correspondence between the maximal left (resp. right) ideals
of A and those of A/N. Likewise there is a one-one cor-
respondence between the maximal left (resp. right) ideals
of B and those of B/rad B. Now rad B = B n N by propo-
sition 1.4, and thus A/N = (B+N)/N =- B/B n N = B/rad B.
Tracing through the mappings, we see the correspondence is
as indicated.

It was in [12] that Ingraham noted that it was possible
to shift the focus of attention from the coefficient ring to
the algebra itself. Rather than attempting to classify in-
ertial coefficient rings, one could ask for criteria for de-
ciding whether or not a given algebra contains an inertial
subalgebra. Since the major result in this paper has a dir-
ect bearing on the present work, we quote it in detail.

A faithful, commutative R-algebra C is said to be a
(G,R)-algebra if it possesses a finite group G of R-auto-
morphisms such that the G-fixed subring of C is R. For

a given maximal ideal m of C we denote
[a E G|o(c)-c E m V c 6 CI

by J .

15

Theorem 1.27: (Ingraham [12]). Let C be a finitely
generated, connected (G,R)-algebra. Then the following
are equivalent:

(a) C contains an inertial R-subalgebra S.

(b) C/n is R-separable and Jm = Jm’ for any two max-

imal ideals m and m' of C.

If (a) and (b) hold, denoting n Jm by I, we have
S = CI and S is a galois extensionmof R (in the sense
of Chase, Harrison, and Rosenberg) with group G/I. Hence
S is finitely generated, projective, and is the unique in-

ertial R-subalgebra of C.

Theorem 1:28; (Ingraham [12]) Let C be a finitely
generated (G,R)-algebra with R connected. Then the fol-
lowing are equivalent:

(a) C contains an inertial R-subalgebra S.

(b) C/n is R-separable and Jm = Jm’ whenever m

and m’ are maximal ideals of C excluding the

same primitive idempotent.

Chapter II
EXISTENCE OF INERTIAL SUBALGEBRAS OF
ALGEBRAS SEPARABLE OVER THEIR CENTERS
Let A be a finitely generated R-algebra which is sep-
arable over its center C. In this chapter we investigate
the relationship between inertial R-subalgebras of A and
inertial R-subalgebras of C. We show that the existence of
an inertial R-subalgebra of A always implies the existence
of an inertial R-subalgebra of C. The main result of this
chapter gives circumstances under which the converse is
true. Viewed with respect to the main result, the chapter
can be roughly subdivided as follows: preliminaries and
necessary conditions, discussion of the setting, proof of

the conjecture, applications, and examples.

Proposition 2.1: Suppose A is a finitely generated R-alge-
bra which is separable over its center C. If A possesses
an inertial R-subalgebra B, then A = BC. Moreover,

A a B ® C under the correspondence Zlbi ® ci €——> Z)bi ci
where SS = Z(B).

Proof: In view of proposition 1.20 we have N = nA

where N = rad A and n = rad C. Then

n-(A/BC) = (nA + BC)/BC = (N + BC)/BC = A/BC,

since B + N = A by hypothesis. Therefore A = BC by the
Original Nakayama Lemma (lemma 1.1).

16

17

Now since A = BC it is clear that S = Z(B) : Z(A) = C.
Thus it makes sense to speak of B 3 C.

Let g: B ® C 4 A be the natural C-module mapping in-
duced by u(b ®Sc) = bc. Now u[(b ® c)(b’ ® c’)] =
u(bb’ ® cc’) = bb’cc’ = bc b’ c’ = u(b ® c)u(b’ G c’), since
C is the center of A. Hence u is a C-algebra homomor-
phism.

It is clear that u maps onto A since u(B g C) = BC =
A. Thus to prove u is an isomorphism we need only show
it is one-one. Since ker u is a two-sided ideal of B S C,

S
there exists an ideal w c C such that fl(B 8 C) = ker u

S
(proposition 1.19). Let a E u = C n ker u. Then

0 = u(a)

Met-1) = au(l) = a-1 a.

Thus m = (0), and therefore ker u = (0). This proves H

is one-one, and the assertion now follows.

Theorem 2.2: Suppose A is a finitely generated R-alg-

ebra which is separable over its center C. If A contains
an inertial R-subalgebra B, then C possesses a unique

inertial R-subalgebra Z(B) = B n C.

gpppf: It follows from proposition 2.1 that C includes
Z(B). Therefore B n C : Z(B) c B n C, or B n C = Z(B).

Being an inertial R-subalgebra of A, B is separable
over R. Therefore Z(B) is R-separable by proposition 1.13.
Thus we have only to show that Z(B) + n = C, where n =

rad C.

18

Now n = C n N by proposition 1.4. Moreover, Z(A/N)==
[C + N]/N by proposition 1.14 since A is C-separable. Like-
wise Z(B/B n N) = [Z(B) + B n N]/B n N since B is R-sep-
arable. With these facts in mind the following chain of

isomorphisms is clear:

[Z(B) + nJ/n e Z(B)/Z(B) n n = Z(B)/Z(B) n B n N a
[Z(B) + B n N]/B n N = Z(B/B n N) a Z([B + N]/N) = Z(A/N)

= [c + N]/N m c/c n N = C/n.

The composite of these isomorphisms is the identity map,
whence Z(B) + n = C. Therefore Z(B) is an inertial R-sub-
algebra of B.

Uniqueness follows easily. Since C is a direct sum-
mand of A (proposition 1.18) and so is finitely generated
over R, inertial R-subalgebras of C will also be finitely
generated over R (Theorem 1.22). That any two finitely
generated inertial subalgebras of a finitely generated and
commutative algebra coincide is a consequence of proposition
2.6 of [12].

Q.E.D.

Let us denote the n-by-n matrices over a ring R by

722,, (R) .

Proposition 2.3: Let C be a finitely generated, commuta-
tive R-algebra. If C contains an inertial R-subalgebra S,

then ‘mn(S) is an inertial R-subalgebra of ‘mn(C) for each

integer n.

19

Proof: 'When we refer to mn(S) as a subalgebra of
mn(C), we actually mean that subalgebra 8 a-r7I(n(S) of ‘mn(C)

defined by 8 = [(cij)|c 6 S]. It is well-known that‘mn(S)

ij
is separable over S with separability idempotent

n
= e. e 0
e .2: 1r ® r1
i=1

for any r s n. Thus 8 is separable over S and hence
separable over R by transitivity (proposition 1.12). That

rad‘mn(c) = [(cij)|c E n = rad C] is theorem 6.15 of [15].

ii
It is therefore clear that S + rad‘mn(C) =‘mn(C). Thus
mn(S) is an inertial R-subalgebra of ‘mn(C).

Let us suppose that C is a finitely generated, commu-

tative R-algebra possessing a finite group G of R-automor-

phisms of C. Then G can be extended to a group G of R-

automorphisms of ‘mn(c) is an obvious way. If [eij] is

the set of n2 matrix units of’ mn(C), we define E'E E

by 3'( Z’Cij e ) = Z)o(cij)e... That 3' is indeed an R-

ij 1)
automorphism.of' mn(C) follows from the fact that the multi-
plication constants of the matrix units are in the G-fixed

subring of C.

Definition 2.4: A finitely generated, faithful R-algebra

A possessing a finite group G of R-automorphisms with G-

fixed subring AG = A is said to be a (G,A,R)-algebra. If

A is a commutative (G,R,R)-algebra, we refer to it as a

(G,R)-algebra (see [12]).

Let A be a (G,A,R)-algebra. We denote by IA the

normal subgroup of G defined by [c e G|o(a)-a 6 N V a 6 A].

20

When there is no confusion as to which (G,A,R)-algebra is
under discussion we will use the abbreviated notation I

for IA‘

We now propose to abstract the setting of proposition
2.3. Let A be a finitely generated R-algebra which is sep-
arable over its center C. Suppose further that C is a

connected (G,R)-algebra. Let us assume for the moment that

A contains an inertial R-subalgebra B. Then A a B S C

S
‘where S = Z(B) by proportion 2.2. It follows from theorem

1.27 that S can be characterized as being the Ic-fixed sub-
I
ring of C, i.e. s = C C. Hence Ic can be extended to a

group I 2 IC of S-automorphisms of A by defining '3 E'I as

9-03 ® C) = b 6?) 0(c), where o e I Let

c'
IA=TEeY|E(a)-aeNVaeA}.

For 0 6 IC we see that

__ ._ r r
o a - = o b. ® c. - b. ® 0. =
I ’ a (3’21 3 3) £1 3 J
r
= b. . - . = .
jEH 3 ® (0(c3) c3) 6 nA N
Therefore 0 6 IA' implying that Ic = I = IA'

We are thus motivated to consider the following setting

which we hereafter refer to as (*):

(*) A is a finitely generated R-algebra which is separable
over its center C. Further, C is a (G,R)-algebra with G
extendable to A in such a way that IA = IC’ We denote

this common subgroup of G by I. The situation is indicat-

ed by the following diagram:

21

P A
u n
A 2 AI 2 AG
C a CI 3 CG
H M
S R

Remark 1: In the terminology of definition 2.4, A is
seen to be a (G,A,R)-algebra.
Remark 2: If C is connected and possesses an inertial

R-subalgebra T, then T = S by theorem 1.27

Proposition 2.5: Let A be an R-algebra in the setting (*).
Suppose C is connected and contains an inertial R-subalge-
bra S. Then T a A S S. Moreover, I is R-separable if
and only if A is R-geparable. When this is the case,

Z(F) = S if and only if Z(A) = R.

Proof: The technique employed is due to DeMeyer in [7],
and this result is a sharper version of his Theorem 3. We

first need to state a definition and to quote a short lemma.

Definition: A (not necessarily commutative) (G,A,R)-

algebra A is said to be a galois extension of A provided
n

there exists [xi,yi] ; A, 1 s i s n such that Z)xio(yi)==
i=1

6 V o E G ‘where

1,0

6 _ { 1 if g = 1
1,0 I 0 if o # l

22

(ppmma:(DeMeyer [7]) Let A and B be R-algebras with
common subalgebra C. Suppose G is a group of R-automor-
phisms of A and of B in such a way that both A and B
are galois extensions of C. If f: A 4 B is a ring homo-
morphism which commutes with G and fixes C, then f is
an isomorphism.

In the setting of proposition 2.5, there exists a nat-
ural ring homomorphism h: A ® S 4 r induced by h(x ® s)==xs.
Now 8 is galois over R wiih group G/I by theorem 1.27,
from which it follows that F is galois over A. Since R
is a direct summand of S by corollary III.1.3 (2) of [8],

A c A ® 8. Let G/I act on A ® S by SIX 0 s) = A ®‘3(s).
R G/I R

Then (A g S) = A by corollary III.1.3 (3) of [8]. Now

since it fixes A and commutes with G/I, h is an isomor-

phism by the above lemma. Therefore F'3 A ® S.

Since R is a direct summand of S, w: conclude from
proposition 1.15 that P is separable over R if and only
if A is separable over R.

Now if A is central separable over R, then T is
central separable over S by proposition 1.15. Conversely,
assume 8 = Z(r). If x E Z(A), then x E Z(F) = S since

r = AS. Therefore x e A n s = CG = R.

Q.E.D.

We will find the so-called "trace map" provides a con-
venient characterization of inertial subalgebras of algebras

satisfying (*).

23

Definition 2 6: Let G be a finite group of automor-

 

phisms of an R-algebra A. The associated trace map 13 of G,
T : A4A=AG, is definedby T (a) =Zo(a) VaEA.
G G
06G
Remark: TG(A) is a two-sided ideal of A = AG. Thus

7G: A 4 A is onto if and only if 1 E TGIAI-

Proposition 2.7: Suppose A is an R-algebra satisfying (*).

Then TI(A) = T if and only if r = III is a unit in R.
Proof: Suppose r = III is a unit in R. Then
71%) =Zo(%) =23 %o(l) =%21=%|1| =1.
oeI 0€I 061
Thus TI(A) = AI = F.

Conversely, let 7(A) = P ‘where we denote TI by T.

We know for o e I that 0(a) - a e N = rad A V a 6 A.

Thus T(a) - ra E N V a E A. If r is not a unit, it lies
in some maximal ideal m of R. Since T(A) = r, there
exists x e A such that T(X) = 1. Then 1 - rx = T(x)-rx

e N, and so 1 e N + mA. This implies m(A/N) = A/N. By
the Generalized Nakayama Lemma (lemma 1.2), this implies that
mJ-annihR(A/N) = R. Since A is R-faithful,annihR(A/N) =
R n N: and since A is a finitely generated R-algebra,
R n N = rad R by proposition 1.4. But m + rad R = R is
a contradiction, since rad R c m’ for all maximal ideals m’
of R. Hence r t m for any maximal ideal m of R, and
therefore is a unit in R.

The following proposition and its corollary illustrate

the usefulness of the trace map.

24

Proposition 2.8: Suppose A is an R-algebra satisfying (*).

If r = III is a unit, then r + N = A.

Proof: By definition of I, 0(a) - a E N V a 6 A and

every 0 e I. Denoting T by T, we have

I

T(a)-ra€NVaeA.

1 1
Hence r T(a) a e N. But r 7(a) - TIr a). Thus

a - T(%a) 6 N V a 6 A.

or T(A) + N = A. Since 7(A) c r in general, this says

I‘+N=A.

Corollary 2.9: Suppose A is an R-algebra satisfying

(*). If III is a unit, then A = CT.

Proof: Let n = red C. Then n. (A/CI‘) =[cr + nA]/CI‘.
Since A is separable over C, nA = N by proposition 1.20.
Thus [CT + “ALCF = A/CT by proposition 2.8, and therefore

CT = A by the Original Nakayama Lemma (lemma 1.1)

Proposition 2.10: Let C be a commutative ring with
rad C = n. Suppose M is a finitely generated and project-

ive C-module. If M/nM is free over C/n with basis

n n . .
[xi + "M]i=l' then {xi]i=l is a free basis for M over
C.

Proof: (see [10], page 43). Let us denote the free
C-module of rank n by C(n). There exists a natural C-

(n)

module homomorphism m: C 4 M defined by

n
mncl.---.cn>] = 2: cix

i=1 i'

25

Since [xi + nM]2=1 span M/nM, we have m(C(n)) + "M = M
or n-M/m(C(n)) = M/m(c(nH. Hence M = w(C(n)) by Nakayama's
Lemma.

Thus the sequence 0 4 ker m 4 C(n) 9 M 4 0 is exact.

Since M is C-projective the sequence splits, and C(n)

ker m G L for some submodule L of CI“). Let (c1,---,c )

n n n
6 ker m. Then Z) c.x. = 0 so that. Z)c.x. + nM = 0, whence
i=1 1 1 i=1 1 1

the coefficients in C/n determined by the C1 are zero
(i.e. ci 6 n V i s n). Therefore ker m : nM = n ker m 9 n L,
whence ker T = n ker m. Since ker m is a direct summand of
M it is finitely generated over C. Thus ker m = (0) by

Nakayama's Lemma.

gorollary 2.11: Suppose A is a finitely generated R-
algebra in the setting (*L and that r = III is a unit in
R. If C is (i) a principal ideal domain or (ii) semi-

local and connected, then A possesses an I-invariant basis.

gppgg: Since r = III is a unit, r + N = A by prop-
osition 2.8. As we have previously noted, A being separ-
able over C implies that N = nA. Therefore any generat-
ing set (resp. basis) for A over C is congruent modulo
nA to an I-invariant generating set (resp. basis) for A
over C. Thus we need only show that A/nA possesses a
finite basis over C/n. This will follow from that fact
that under either condition (1) or condition (ii) A pos—
sesses a finite basis over C.

(i) Suppose C is a principal ideal domain. Then it

is well-known that any finitely generated C-module can be

26

represented as a direct sum of a free C-module of finite rank
and its torsion submodule: A a C(n) @>t(A). However, since
A is projective over C, t(A) =(0).

(ii) Suppose C is connected and semi-local. It fol-
lows from corollary 1.9 that A is free of finite rank over
C.

We now prove a preliminary version of our main result.

Lemma 2.12: Suppose R is a semi-local ring and A
is a finitely generated R-algebra in the setting (*). Assume
that r = III is a unit in R. If C contains an inertial
R-subalgebra S and C/n is R-projective, then T is an

inertial R-subalgebra of A and r'm A/N.

Proof: Since r - III is a unit, I + N = A by prop-
osition 2.8. C is finitely generated over R since it is
a direct summand of A (proposition 1.18), and thus is seen

to be semi-local. Therefore A possesses an I-invariant

n

baSis {xi}i=1

over C by corollary 2.11. This in turn

implies r n N

(0). To see this we suppose Y E T n N.
n
Then, since N = nA, y has a representation as y==f3uixi
i=1

‘where “i E n. Moreover, since y E F, 0(y) = y V o 6 I.
n n

Thus .2 0(ui)xi =.Z) uixi. Hence ”i = (Hui) V o e I. There-
i=1 Ii=1

fore “i E n n C n n S. However. n n S = (0) by Lemma

2.3 [12],since C/n is R-projective. Thus “i = 0 Vi.
whence P n N = (0).
Now C/n is necessarily R-separable since it is a homo-

morphic image of the separable R-algebra S:

c/n = (S+n)/n * S/S n n.

27

That A/N is R-separable follows from the transitivity of
separability (proposition 1.12) since A/N = A/nA is cen-

tral—separable over C/n. Hence

Fer/(0) =I‘/I‘nN=:[I‘+ N]AT=A/N

is R-separable and so is an inertial R-subalgebra of A.

We note that in this instance we have a "classical"
Wedderburn decomposition of A in the sense that A = T'® N.
just as in the case where A is an algebra over a field.

We are now in a position to prove the main result of

this chapter.

Theorem 2.13: Let A be a finitely generated (G,A,R)-
algebra in the (*) setting. Suppose A is separable over
its center C which is assumed to be connected. If r = III
is a unit and C contains an inertial R-subalgebma S, then
T = AI is an inertial R-subalgebra of A. Furthermore

A a T ® C a A 8 C.
S R

Proof: A = r + N by proposition 2.8, since r = III
is a unit. However, to show r is R-separable is more

challenging.

Idea of the proof: Let m be a maximal ideal of S.
We will show A/mA is an S/m-algebra which fulfills the
hypotheses of the previous lemma.and so possesses an inertial
S/m-subalgebra. We will further demonstrate that this iner-
tial S/m-subalgebra is in fact T/mP- By the theorem of
Endo‘Watanabe theorem 1.2D, we conclude r is separable

over 8. Since S is separable over R, r is separable

over R by transitivity (proposition 1.12).

28

As a first step we show that A/mA is an S/m-algebra
satisfying the setting (*).

(l) A/mA is a central separable C/mC-algebra.

Since A is separable over C, A/mA is separable over
Z(A/mA) =[C + mALfiA a C/C n mA by proposition 1.14. Now
mC is a two—sided ideal of C, and so C n mA = C n (mC)A =
mC by proposithan 1.19.

(2) A/mA possesses a group I. of S/m-automorphisms
‘which restricts faithfully to C/mC (i.e., the restriction
of II to C/mC is a group of S/m-automorphisms of C/mC
'which is isomorphic to I).

Since m c S c CI, it follows that 0(mA) = mA for ev-

ery o E I. Thus each 0 e I induces a well-defined mapping

on A/mA by o(a+mA)

0(a) + mA. It is clear that this

mapping is an S/m-algebra automorphism. Let

JmA=[oeIIo(a)-a€mAVaEA].

One easily sees that JmA 9 I and that I/JmA acts faith-

fully on A/mA.

Now by hypothesis I also acts as a group of S-automor-
phisms on C. Therefore I/Jmc
S/m-automorphisms on C/mC where

acts as a faithful group of

Jmc= (o e IIo(c)-c e mCVc e c}.

We now assert that JmC = JmA' one consequence of which

is that I/J acts on A/mA and restricts faithfully to

mA

C/mC. Since mC = C n mA. it is immediate that JmA : JmC'

Let 0 E Jmc. By corollary 2.9 each a e A has a representa-

29

n
tion as a = Z)c.x. where c. e C and x. e r. Thus
i=1 i i i i

E) n n
o(a)-a = o( c.x.) - Z)c.x. = Z)(o(c.)-c.)x. 6 mCP = mA,

i=1 i 1 i=1 i i i=1 i i i
shoWing o 6 JmA' from which we conclude JmA = JmC' ‘We de-
note I J = I J b -E

/ mA / mC y I

(3) (AMA): =[I‘+mAMuA and (c/mc)I =[s+mCJ,fic.

Clearly Ir+mALfiA c (A/mA)I. To prove the opposite inclu-

sion, let 3'6 (A/mA)I. Since III is a unit, the argument

given in proposition 2.7 shows 3 ;.€ A/mA such that
3:?

Now let {Oi}§=1 be a full set of coset representatives for

. I. '. , = ' ° ... .
JmA in p (i e I (disjoint) JmA U JmAUZU U JmAOP)
Thus a =iEH oi(x) + mA. Let n = IJfiAI and b =i=loi(x)
so that b = a. Claim nb - TI(X) 6 mA. For
P P
nb-TI(x) =n 23010:) - Zo(x) =n Zoi(x) -
i=1 061 1:1

P P P
- Z) ( Z)eoi(X)) = n .710i(XI - Z)( Z) go (X)) =
1:

. . l
= = f
geJ p i l i l ,eJ P

p
= Z) [ ZE(Oi(XI - :Ui(X))] 6 mA
geJmA i=1

since

oi(x) - 501(X) 6 mA

by the definition of J Thus na'= TI(X), or

mA.

 

HI") = 71% x) EIF+mAYmA-

one

a:

30

An identical argument shows (C/mC)I =IS+mdymC.

(4) C/mC is a connected (I) S/m)-algebra.

That C/mC is connected follows from proposition 1.26
which states that there is a one-one correspondence between
maximal ideals m c S and maximal ideals m c C. Since this
correspondence is given by m 4 m + n =‘M 4‘m n S = m where
n = rad C, it follows that there is exactly one maximal i-
deal lying over mC: namely m + n. Thus C/mC is local,
and hence connected. Now (C/mC)T.=[S+mCth a 8/8 n mC by
(3), and we assert that S n mC = m. Since S n m0 is a
two-sided ideal of S which includes m. it must equal eith-
er m or S. Now if S 0 ac = S, then 1 E mC so that
mC = C. This implies that S = m + annihS(C) by the Gener-
alized Nakayama Lemma. However, C is faithful over S and
this leads to the contradiction that m = S. Therefore
3 n me = m. and so (C/mC)T = 5/1".

(5) S/m is an S/m—inertial subalgebra of c/mc.

Rad (C/mC) = fl m/mC where the intersection is taken over
all maximal ideals m 2 m0. Clearly then, rad(C/mC) :
(n+mC)/mc. Moreover, C n C/mC is an epimorphism and so

(n+mC)/mC : rad(C/mC). Hence

S/m + rad(C/mc) = (S+mC)/mC + (n+mCI/mC = (S+n+mC)/mC = C/mC:

proving the assertion.
(6) A/mA is a finitely generated S/m-algebra in the
setting (*). An argument identical to (5) shows rad(A/mA) =

(N+mA)/mA. Hence IA/mA = {3 e TIRE-'5 e rad (A/mAH = Y.

31

Similarly, IC/mc = I, whence I I. In light

=I =
A/mA C/mc
of what we have previously shown, A/mA is in the setting (*).

A/mA 2 (A/NA)I

C/mC 2 (C/mC)I

S/m

(7) IT+mALbA “ T/T n mA is an S/m-inertial subalgebra
of A/mA. S/m is a field and therefore semi-local. Trivial-
ly; (C/mC)/rad(C/mC) is S/m-projective since all modules
over a field are free. Therefore the assertion follows from

Lemma 2.12 and (3).

(8) T 0 nA = (rad S)?

Since A = er and n = rad C, it follows that nA = nr.

q
Therefore each x e r n nA can be represented as x = Z)n.x.

j=1 3 3
where nj e n and xj 6 P. Thus
q q
HRI=ZNR)=ZMZMRJ= Z(ZNmMJ=
oeI oeI j=1 J 3 j=1 061 3 3
q
= Z)TI(n-)X. 6 (S n n)? = (rad S)T.
j=1 J 3
Moreover since x e r, T (x) = Z) o(x) = 22x = rx where
I I
1 gel 061
r = III. Thus x =-; TI(X) E (rad S)r, and therefore

F n nA : (rad S)?. The opposite inclusion holds since rad S =

S 0 n. and so equality is attained.

(9) r n mA = m?
Arguing as above. mA = m(TC) = m (r(n+S)) = m? + mrn :

m? + n? (Recall m c S c T). Therefore each y e r 0 mA

32
k
can be represented as y =:Z) uixi + a where ui E m. xi 6 T

i=1
and a 6 hr. Inasmuch as m? c r, ‘we conclude that

y — £53 uixi E T.

Therefore a E T n n? = (rad S)? by (8). Since rad S c m
it follows that a, and therefore y, is an element of m?-
The opposite inclusion is immediate, and so r n mA = m?-

With these technicalities behind us, it is straightfor-
‘ward to verify that r is an inertial R-subalgebra of A.
Combining (7) and (9), we conclude that F/mT is a separable
S/m-algebra. Furthermore, since A is finitely generated
over 5, it follows that T = TI(A) is finitely generated
over S. Therefore, by the well-known result of Endo and
Watanabe (Theorem 1.21), it follows that r is a separable
S-algebra. Now since S is a separable R-algebra, T is
also a separable R-algebra by the transitivity of separabil-
ity (theorem 1.12). Inasmuch as we have previously shown
that r + N = A, ‘we conclude that r is an inertial R-sub—
algebra of A.

The isomorphism A a T G C is a consequence of propo-
sition 2.1. Moreover, that Sr a A g S follows from propo-

sition 2.5. Therefore, since S is clearly an R-S bimodule,

‘we have A a r ® C a (A 8 S) 8 C a A 8 (S 8 C) a A 8 C.

S R S R S R
Q.E.D.
In the case where r = III is a unit in R, ‘we have the

following extension of Ingraham's theorem 1.27.

33

cgzgllary 2.14: Suppose A is an R-algebra satisfying

(*). Assume that IA = IC and that r = III is a unit in
R. Then A possesses an inertial R—subalgebra r if and
only if A/N is R-separable and Jm = Jfi, for any two max-

imal ideals m and m' of C.

Remark 2.15: In the proof of theorem 2.13 the hypothesis
of A being central separable over C was used only to con-
clude that rad A = (rad C)A. This "lifting of the radical"
property is true in the more general setting where A is
separable and projective over C : Z(A) (see lemma 1.3 (c)
and (d)). Thus we may relax (*) to this new setting and still
Obtain the same conclusions from theorem 2.13.

In the above context, theorem 2.13 is seen to be an ex-

tension of the following theorem due to F.R. DeMeyer.

Theorem 2.16: (DeMeyer [7]) Let A be a finitely gen-
erated, separable, and projective algebra over a commutative
ring K. Assume that K is connected. Suppose G is a
finite group of ring automorphisms of A ‘which restricts
faithfully to a group of ring automorphisms of K. Let R =

G

K and assume that K is a finitely generated, separable,

and projective R-algebra. Then B = AG is R-separable and
A m B ® K. Moreover, if A is central over K, then B
is ceniral over R.

We proceed to show that this theorem is indeed a spec-
ial case theorem 2.13. Suppose A is any K-algebra satis-

fying the hypothesis of theorem 2.16. Then K is clearly

34

a connected (G,R)-algebra, and is an inertial R-subalgebra
of itself. Therefore K = KIC by theorem 1.27, and so

IC = <1). This trivially implies that Ic = IA and that
III is a unit in R. Therefore A also satisfies the
hypotheses of theorem 2.13.

We should point out that we use DeMeyer's techniques in

proposition 2.5 in order to obtain the isomorphism r a A ® 8.
R

Using the techniques introduced [12] it is possible to
relax the connectedness hypothesis of theorem 2.13 at the
expense of a somewhat weakened conclusion.

Suppose R is a connected ring and C is a (G,R)-alge—
bra. It is shown in lemma 2.14 of [12] that C possesses
a finite set [e532]:1 of primitive idempotents such that
C = @.§: Cei. Let e be one of the primitive idempotents

i=1
and define He = [o e IIo(e) = e}. It follows from the proof
of theorem 2.15 of [12] that Ce is a connected (HE,Se)-

algebra.

Iheorem 2.12: Let R be a connected ring and suppose
A is a finitely generated (G,A,R)-algebra which is in the
(*) setting. Assume further that A is separable over its
center C and that r = III is a unit. If C possesses
an inertial R-subalgebra, then A possesses an inertial S-
subalgebra where S = CI.

2599;: That Ae is central separable over Ce follows

easily from the fact that A is central separable over C

and the fact that the projection mapping T: A 4 Ae is a

ring homomorphism.

35

‘We now show that He restricts faithfully from Ae to

Ce. It is a consequence of corollary 2.9 that A = rC ‘where

n
r = AI. Thus each a 6 A can be represented as a = Ecixi
i=1

where ci e C and xi 6 P. Now suppose o 6 Ha restricts
to the identity map on Ce. Then o(ae) = U[( Zlcixi)e] =
n n .=1

230(c.)x.e = Z)c.x.e = as for all ae E Ae, ‘whence o is
i=1 1 1 i=1 1 1
the identity map on Ae.

It is immediate that IAe = [o e HeI(o(a)-a)e e rad(Ae) =

Ne V a E A] = [o e HeI(o(c)-c)e e rad(Ce) = ne V c E C] = ICe

Finally, we show that Se is an inertial Se-subalgebra
of Ce. NOw it is a further consequence of theorem 2.15 of
[12] that the inertial Re-subalgebra of Ce can be charac-
terized as (Ce)Fe where Fe = [oEGIo(e)=e and
o(ce)-ce 6 ne VC 6 C]. Clearly HE = [o 6 GIo(e) = e and

F

e 2 (Ce) e.

o(c)-c 6 n V c 6 C] : Fe' so that Se = (Ce)
Therefore it follows that Se + ne = Ce, whence Se is an
inertial Se-algebra of Ce.

Having verified that Ae fulfills all the hypotheses
of theorem 2.13, we can conclude that (Ae)He is an iner-
tial Se-subalgebra of Ae. It follows that B = 9£§3(Aei)Hei
is an inertial S-subalgebra of A. _

Q.E.D.

Remark: In general one can conclude only that S con-
tains the inertial R-subalgebra of C. In the event equal-
ity is obtained, B will be R-separable by transitivity
and therefore will be an inertial R-subalgebra of A. Such

equality is, of course, attained when C is connected, in

which case the setting reverts to that of theorem 2.13.

36

We now present two examples which provide additional
insight into the implications and limitations of the hypoth-
eses of theorem 2.13.

Suppose A is a finitely generated R-algebra which is
_separable over its center C. If C is a connected (G,R)-
algebra and A contains an inertial R-subalgebra B, then
it is always possible to extend I to R-automorphisms of

C

A in such a way that IC = IA (page 2(1). As the follow—
ing example points out, it is by no means true that every
extension of IC to R-automorphisms of A satisfies IC =

IA‘ We first need to state a definition.

Definition 2.18: An algebra A over a commutative ring

C is said to be a generalized quaternion algebra provided
A is a free C-module having basis [1 6 C, a, 8, a8] ‘with
multiplication induced by a2 = x e C, 82 = y e C, and
8a = -a8. We will denote this algebra by (C,x,y).
ggggplegg.l9: Let A = (C,-l,-1) ‘where C = R[x]/(x2)
and R is the field of real numbers. Let G = (d) be the
cyclic group of R-automorphisms of C of order tWO‘With gen-
erator a defined by o(r + s;) = r - s}; Then A is sep-
arable over C, C is a connected (G,R)-algebra, and A
possesses an inertia1.mrsubalgebra. Furthermore, there ex-
ists an extension ‘3 of G to A in such a way that
IC g IA.
Discussion: C is local with unique maximal ideal
m = (x)/(x2). One sees A/mA a (R,-l,-l), which is the clas-

sical quaternion algebra. (R,-l,-1) is well known to be

37

(central) separable over R a C/m. Therefore A is separ-
able over C by Endo‘Watanabe (Theorem 1.21). One easily
checks that A is,in fact.central separable over C.

Now R is clearly an.R-inertial subalgebra of C, and
it is immediate that C is a (G,R)-algebra. Since rad A =
mA, it follows easily that.(R,-1rl) is an inertial R-sub-
algebra of A.

We define 3: A 4 A by 3(c1-l + cza + C38 + c4oB) =
o(cl)-1 + 0(c3)a + 0(c2)8 - 0(c4)a8. It is a straightforward
verification to show 3 is an R-automorphism of A ‘which
has order two. Thus ‘3 = (3) is an extension of G to A.
New since A is central separable over C, it follows that
rad A = 773A = 772-1 o m-a 9 me 63 man. Since o(r+s§) - (r+s§:') =
2s; 5 m, we see that '3 = IC' However, 31a)-a = 3(c1-l +
cza + C38 + c4a8) - (cl-1 + cza + c38 + c4a8) = (o(cl)-cl)'1 +
(C(c3)-c2)a + (0(c2)-c3)8 - (0(c4)+c4)a8. Thus for an element
a E A such that 0(c2)-c3 é m (e.g. c2 = %- and c3 = ~%),
o(a)-a é rad A. Therefore IA = <l>, and we have demonstra-
ted that C = IC 2 IA.

We remark that 5': <E>, where '3(c1°1 + cza + C36 +
c4oB) = o(cl)'l + 0(c2)a + 0(c3)8 + 0(c4)a8, is an extension

of G to R-automorphisms of A in such a way that IA = IC.

As shown by the following example, the hypothesis that
r = III is a unit is not necessary for the existence of an
inertial R-subalgebra of an algebra A in the setting (*).
Example 2.29: Let A = C = Z[x]/(x2), and let G be

the cyclic group of Z-automorphisms of C of order 2 with

38

generator 0 where o(a +‘b;) = a - b;. Then C is a con-
nected (G,Z)-algebra with inertial Z-subalgebra S = Z.
However III is not a unit in Z.

Discussion: Although C is not local, a straight for-
ward calculation shows it is connected. As in example 2.19,
n = rad C = (x)/(x2). From this it is clear that Z adds
with rad C and so is a Z-inertial subalgebra of C. One eas-
ily checks that I = [o e GIo(c)-c e n V c E C] = G. There-
fore the order of I is 2 which is not a unit in Z.

Settings where the order of a group of automorphisms
is a unit have precedent in the literature. For example, Y.
Takeuchi shows in [18] that if a ring r is a galois exten-
sion of its center C with group G, then n = IGI is a unit
in C. A related result was obtained by T. Kanzaki in [13]
where he proved that if r is galois over R ‘with group G

and H = [o 6 GI 0 IC = 1], then r = IHI is a unit in R.

Chapter III

UNIQUENESS OF INERTIAL SUBALGEBRAS OF
ALGEBRAS SEPARABLE OVER THEIR CENTERS

The main objective of this chapter is to investigate
the question of uniqueness (up to isomorphism) of inertial
R-subalgebras of those finitely generated R-algebras which
are separable over their centers. Suppose C is a finitely
generated, commutative R-algebra with inertial R-subalgebra
S. Let us denote the Brauer groups (definition 3.1) of S
and of C by 8(5) and 8(C) respectively. ‘We show that
under certain conditions the inertial R-subalgebras of any
central separable C-algebra are inner automorphic if and
only if the induced mapping 8(8) 4 3(C) is a monomorphism.
We conclude by presenting the only known example of an R-
algebra which possesses non—isomorphic inertial R-subalgebras.
The example will be seen to be a member of the class of alge-
bras studied in the previous chapter.

In 1960 Auslander and Goldman introduced the concept of
the Brauer group of a commutative ring R. The Brauer group
of R is an abelian group which reflects the variety of cen-
tral separable R-algebras. It is also an important invari—
ant of the ring R.

Let 0(R) denote the set of isomorphism classes of cen-
tral separable R-algebras. Let o°(R) denote the subset

of 0(R) consisting of those classes each of whose elements

39

40

is isomorphic to the endomorphism ring of a finitely gener-
ated, projective, and faithful R-module. Elements A, B e 0(R)
are said to be equivalent (At~ B) provided there exists
elements X, Y E 0°(R) such that A ® X a B ® Y.

Definition 3. : Let $(R) denofie the sit of equiva-
lence classes of 0(R) under the equivalence relation
For arbitrary [A], [B] E m(R), the binary operation given
by [A][B] = [A ® B] is well-defined and makes $(R) into

R
an abelian group called the aging; group of R ([8], page 60).

The identity element of 8(R) is the class [R] containing
the ground ring. The inverse of any class [A] is the class
[A°] containing the opposite algebra A° of A.

Suppose f: R 4 S is any ring homomorphism of R to
the commutative ring S. Then f endows S with the struc-
ture of an R-algebra, and fi(f): m(R) 4 3(8) defined by
$(f) [A] = [A ® 8] is a homomorphism of the Brauer groups.
Indeed, one segs that s( ) is a covariant functor from
the category of commutative rings (and ring homomorphisms)
to the category of abelian groups (and group homomorphisms).

Definition 3.2: The Brauer group $(R) of a connected
ring R is said to have the unigue representation properpy
provided each class [A] e 8(R) has a representative D
possessing no idempotents other than 0 and l and having
the property that to each B E [A] there corresponds an in-
teger n such that B aumn(D).

Remark 3.2.1: The representative D of the class [A]

is unique up to isomorphism. Suppose D* is another repre-

41

sentative of [A] having the same properties as D. Then
there is an integer r such that mr(D*) a D. Since D
has no idempotents except 0 and 1, it follows that r = l.
whence D* a D.

Remagk Bigpl: The integer n associated with each
8 e [A] is also unique. Suppose 'mn(D) a‘mr(D). Then
mn(D) g Rm m'mr(D) g Rm for every maximal ideal m of R,
or equivalently' mn(Dm) m‘mr(Dm) for every maximal ideal m
of R. Since Dm is free of finite rank over Rm by prop-

osition 1.8, it follows that n2[Dm:Rm] = r2

[szRm] for
every maximal ideal m of R. Therefore n = r.

It is well-known that if F is a field then $(F) has
the unique representation property. In this instance the
representative D of each class [A] e $(F) is seen to be
a division ring. F. R. DeMeyer provides us with the follow-
ing generalization of this result ([6], corollary 1):

Theorem 3.3: If K is a connected, semi-local ring,
then 8(K) has the unique representation property.

R. Hoobler in his dissertation [11] extends the defi—
nition of the Brauer group from fields to commutative rings
by introducing a stronger equivalence relation ~’ on 0(R)
than did Auslander and Goldman. Central separable R—algebras
A, B E 0(R) are said to be ~’—equivalent (A ~’ 8) pro—
vided there exists integers u and t such that mu(A) a

mt(B). Let J(R) denote the set of isomorphism classes of

finitely generated, faithful, and projective R-modules.

42

o(R)/~’ forms an abelian group,which is denoted '§(RL under
the operation [A] [B] = [A 8 B]. This is a consequence of
a result by Bass [3] which siates that to each P E J(R)

there corresponds a Q 6 J(R) and an integer n such that
P 8 Q a.@ .E) R. One immediately sees that there is a nat-

R i=1 ._
ural epimorphism from 8(R) onto 8(R).

Definition 3.4: The uniqueness statement is said to
hold for a finitely generated R—algebra A provided any two
inertial R-subalgebras B and B’ of A are isomorphic
via an inner automorphism of A generated by an element

of the form 1 - n, for some n 6 N = rad A.

Lemma 3.5: Let C be a finitely generated, commutative
R-algebra with inertial R-subalgebra 8. Suppose W is a
central separable S-algebra. Then W' is an inertial R-sub-
algebra of ‘W ® C.

.gpggg: Since ‘W is projective over S (proposition
1.17), W can be considered an R-subalgebra of ‘W 8 C by

8
identifying it with ‘W ® 8. That W is R-separable follows

S
from the transitivity of separability (proposition 1.12).

Now rad (W B C) = n-(W ® C) by proposition 1.20, and so we
S

S
have
w + rad(w e C) =‘W B S + W e n = W e (S+n) = W G C
8 S 8 S 8
Therefore W’ is an inertial R-subalgebra of W 8 C.
8

Theorem 3.6: Let C be a finitely generated, commuta-
tive R-algebra possessing an inertial R-subalgebra S.

(a) Suppose $(C) = ETC). If the uniqueness statement
holds for every central separable C-algebra then 8(8) 3 8(C)

is a monomorphism.

43

(b) Suppose C is connected,and both 8(8) and 8(C)
have the unique representation property. If 8(8) 3 8(C)
is a monomorphism, then the inertial R-subalgebras of any
central separable C-algebra are isomorphic as S-algebras.
2529;: (a) Suppose [B] E ker n so that [B 8 C] =
[C] in 8(C). Since 8(C) = 8V0), there exist iitegers

n and m such that (B ® C) 8 mn(C) =- mm(C). Now (B 8 C)

S C S
®77zn(C) =- B ®(C emn(C)) =~ B a (mum) =~ a s (722nm s e) =-
c . s c s s s
mn(B) 8 C. Since both E and ‘mn(8) are central separable

S

over. S, mn(B) m B g‘mn(8) is central separable over 8
also. Hence mn(B) is an inertial R-subalgebra of *mm(C)

by lemma 3.5. However, Imm(8) is also an inertial R-subalg-
ebra of mm (C) by proposition 2.4. Thus 8 gmnw) =- 771m(8)
by hypothesis, whence B ~o8 in 8(8). Therefore ker n =
(0), or, equivalently, 8(8) 3 8(C) is a monomorphism.

(b) Let A be any central separable C-algebra and sup—
pose that B and B’ are inertial R-subalgebras of A. Then
A a 8 8 C w 8’ ® C by proposition 2.2. Hence [8 8 C] =
[B’ 8 g] in 8iC). Now 8(8) 3 8(C) is a monomoiphism by
hypoihesis,so that [B] = [B’] in 8(8). Since 8(8) has
the unique representation property, there exists integers r
and t and a representative D 6 [B] possessing no idempo-
tents except 0 and 1 such that B e-mr(D) and B’ m

S
mr(D ® C) ermt(D 8 C). Since D 8 C is a central separable

S S S
C-algebra,and since 8(C) has the unique representation prop-

mt(D). It follows that ‘mr(D) ® C =3-:7)It(D) ® C, 'whence
S

erty, there exists an integer u and a representative

44

3 E [D 8 C] containing no idempotents except 0 and l
S
and and! that D ® C °' mu (.3) -Thus Wtrmuwl) a: mt (mu (19)) or.

equivalently, mriffil almtuIflI' The dimension of such mat—
rices is uniquely determined (remark 3.2.2), and so r = t.
Therefore B a B’ as S-algebras.

Q.E.D.
Proposition 3.7: Suppose A is a finitely generated R-alg-
ebra containing an inertial R-subalgebra B. Then every cen-
tral idempotent of A is contained in B.

Proof: Any central idempotent e e A induces a ring

direct sum decomposition of A in the usual way: A = Ae ®

A(l-e). The projection map w: A 4 Ae defined by w(a) as

is thus seen to be a ring epimorphism. Therefore w(B) Be

is R-separable by proposition 1.14. Likewise, B(l-e) is
R-separable. Therefore the R-subalgebra B’ of A defined
by B’ = Be @ B(l-e) is R-separable, since it is the ring
direct sum of two separable R—algebras. It is clear that
B’ a B, whence B’ + N'= A. Thus B’ is an inertial R-
subalgebra of A. Now both 8 and B’ are finitely gener-
ated by theorem 1.22. Since any two nested and finitely gen-
erated inertial R-subalgebras coincide ([12], lemma 2.5),
it follows that B = 8’. Therefore e is an element of 8.
Remark: Non-central idempotents of a finitely gener-
ated R-algebra need not be contained in a particular inertial
R-subalgebra. This fact is nicely illustrated in example 3.9
‘where A ==m2(C) possesses many non-central idempotents,

none of which are in the inertial R-subalgebra B = (8,-l,-l).

45

Theorem 3.8: Suppose R is a semi-local ring and C
is a finitely generated, commutative R-algebra with inertial
R-subalgebra S. Then 8(8) 3 8(C) is a monomorphism if
and only if the uniqueness statement holds for every R’
algebra which is central separable over C.

gpggf: Since both C and 8 are finitely generated
over R, they also are semi-local. Any semi-local ring can i

be decomposed into a direct sum of connected semi-local rings:

n E

C = 6 Z) Cei where [ei]n is a set of primitive ortho- i
1:1 i=1 n 5

gonal idempotents such that l = ‘ZIei. Since [ei] : S by I
i=1 .

 

If-
Reg‘ '-

proposition 3.7, 8 can also be decomposed as a direct sum
of its ideals generated by the same primitive idempotents:

S = @ .EI Sei. For the sake of convenience, let us denote
Cei by=lci and Sei by Si' It is seen that Si is an
inertial R-subalgebra of Ci‘ As indicated in the proof of
theorem 2.17, a decomposition of C induces a decomposition
on any C-algebra A in such a way that A is (central)
separable over C if and only if A1 is (central) separable
over Ci for all i s n.

With this is mind, we reduce the theorem to the case
where C is connected.

(a) The uniqueness statement holds for every central
separable C-algebra if and only if it holds for every central
separable Ci-algebra for every i S n.

Suppose the uniqueness statement holds for every central

separable Ci-algebra for every i s n. Let B and B’ be

inertial R-subalgebras of an arbitrary central separable C-

46

algebra A. Then Bi and Bi are inertial R-subalgebras
of the central separable Ci-algebra Ai. Therefore there
exists an inner automorphism 9i of A1 such that 91(Bi) =

I _ ... _ =
Bi , 'where 91(ai) — (ei n{)ai(ei ni) and ni,n§ 6 rad(Ai)
N. .
:11

Z)ni) is an inner automorphism of A generated by an ele-
i=1
ment in N = rad A, and is such that 9(B) = 8’.

Then 9: A 4 A defined by 9(a) = (1 - fl n{)a(l -
i=1

Conversely, assume the uniqueness statement holds for
all central separable C-algebras. Let i be any integer s n,
and let Bi and Bi be inertial R-subalgebras of an arbi-
trary central separable Ci-algebra Ai. One sees that

_ 0.. 0.. ’= .0. .
13-319 @Si-l®81®si+l® sash and B sle eskle

Bi @ Si+1@---@ 8n are inertial R-subalgebras of the central

separable C-algebra A = C1®---@Ci_l 6 Ai Q'Ci+1®°"® Cn'
Therefore there exists an inner automorphism 9 of A such
that 9(8) = B’ where 9(a) = (1-n*)a(1-n) and n,n* E N =
' ' = - 1" .
rad A. Then 91. Ai 4 Ai defined by 91(ai) (ei n1)a1
(ei-ni) is an inner automorphism of A1 generated by an
. __ - _ I
element in N1 — rad (A1), and is such that 91(81) — Bi’
(b) 8(8) 3 8(C) is a monomorphism if and only if each

n.
8(Si) 41 8(Ci) is a monomorphism for every i s n.

Every central separable S-algebra B can be decomposed

n n
as B = e 2- Bi where Bi = Bei. Thus B e c = (of) Bi) :8
n i=1 n'n) S 1:]. S
((92 c.) =- e B. o c.. Now B. e c. = (0) for i y! j.
j=13 (i.j)1s 3 1s 3

since the set of idempotents [ei] is orthogonal. Also

Bi ® Ci = Bi 8 Ci' Since @JZ)Sj c annihS(Ci). Therefore
8 S. If]
i
5 i=1 1 Si

47

It is well-known that the Brauer group distributes over

n
finite direct sums. Hence we may identify 8(© Z) Ci) with

n n n 1:].
e 238(Ci). and Mo 2381) with 928(81). Thus nIB] =
1:1 i=1 i=1
n n n ]
[B®C]=I®Z B.®C.]=®Z[B.®C.]=@7)n.[B..
8 i=1 1 si 1 i=1 1 si 1 i=1 1 1

Now suppose that ”i is a monomorphism for every i s n,
and let n[B] = [C]. Then ni[Bi] = [Ci] for every i s n,
whence [Bi] = [Si] for every i s n by assumption. This
implies that [B] = [8], and therefore that n is a mono-
morphism.

Conversely, assume n is a monomorphism. Suppose
ni[Bi] = [Ci] for any i s n. It is clear that

’— .00 ea.
8 _ slo 931-1 6) Bi o siflo e sn

is a central separable S-algebra, and that n[B] = [B ® C] =
S
@ Z) n.[B{] = [C]. Then [8] = [S] by assumption, whence

= [8.]. Therefore ”1 is a monomorphism for every

In view of (a) and (b), it suffices to prove the theorem
in the case where C is connected and semi-local. In this
setting if M is a finitely generated projective C-module,
then M is free of finite rank over C (corollary 1.9).

This implies that 81C) = 8(C). Therefore the uniqueness
statement for all central separable c-algebras implies 8(8) 3
8(C) is a monomorphism by theorem 3.6 (a).

Conversely, assume 8(8) 1 8(C) is a monomorphism.

Since 8 and C are connected, semi-local rings, both 8(8)

and 8(C) have the unique representation property (theorem

48

3.3). Therefore the inertial R-subalgebras of every central
separable C-algebra A are isomorphic as S-algebras by
theorem 3.6 (b). we are done if we can show that,in fact,
they are isomorphic via an inner automorphism of A gener-
ated by l - n 'where n e N = rad A. First we need the
following generalization of the Skolem-Noether Theorem which
appears in [17].

Theorem (Sridharan): Let R be a semi-local ring. Sup—
pose A and B are central separable R-algebras with R-
algebra monomorphisms f,g: B 4 A. Then there exists an in-
ner automorphism 9 of A such that g = 9 o f.

Suppose that f: B 4 B’ is an S-isomorphism of two in-
ertial S-algebras B and B’ of a central separable c-alg-
ebra A. It then follows that the mapping f O l: B 8 C 4
8’ ® C induced by f O 1(b8c) = f(b)8c is a C-isomoiphism.

S
In view of proposition 2.1, the multiplication maps )1: B 8C4

S
A and u’: B’ 8 C 4 A are also C-isomorphisms. Then u
S
and “’0 f®l are C-algebra isomorphisms which map B 8 C

S
onto A. Therefore there exists an inner automorphism 9

of A such that u = 9 o [u'o f®l] by Sridharan's theorem.
This implies that B = “(B ® 8) = 9(u’(f®l(B ® 8))) =
6(u’(B’ ® 8)) = 9(B’). S S

Thesfollowing argument, due to Azumaya [2], shows that,
moreover, B is conjugate to 8’ via an inner automorphism
generated by l - n, ‘where n e N = rad A. Suppose 9(x) =

w x w-l. Since A = B + N, we can represent w as w=v+n

where v e B and n e N. The element v is a unit modulo

49

N, and is therefore outside every maximal left and every
maximal right ideal of A. It is thus both left and right
invertible and therefore is a unit. Hence wv-1 e 1 mod N,

and (WV-l) B(wvml’r'1 = w(v-]'Bv)w-1 1

= wa- = 9(B) = B’.
Therefore the uniqueness statement holds for A.
Q.E.D.

We conclude with an example of an R-algebra which has
non-isomorphic inertial R-subalgebras. This example will
be seen to be a member of the class of algebras discussed in
Chapter II. Let us denote the localization of the integers
Z at the maximal ideal (P) by Z .

P

Example 3.9: Let R = S = Z5 and C = Z5 5 i
2

: Zs[i], where i = -1. Let A = (C,-l,-l) be the gener-

@ SZ

alized quaternion algebra with basis [l,a,8,a8]. Then
B = (S,-l,-l) and B’ = m2(s) are two non-isomorphic R-
inertial subalgebras of A.

Discussion: (1) A a-m2(C) (and so is central separ-
able over C). C is local with unique maximal ideal. m =
52 O 52

5 5
a (Z/(S),-l,-l). Since Z/(S) possesses a root of unity

i . It is not hard to see that A/MA =- (C/m,-l,-l)

(namely 2), it follows from page 18 of [9] that

re11 e1; F1/2 (1- int) 1/2 (B-ian

 

 

 

 

e2l e22 l/2(-B-ia8) l/2(1+ia)

-A
h. -‘

forms a matrix basis for A/mA over C/m. Thus A/mA m‘m2(C/m)

SO

and so is separable over C/m. Therefore A is separable
over C by Theorem 1.21 (Endo‘Watanabe).

It is a straightforward computation to show Z(A) = C.
Let y = x + ya + 28 + woB e Z(A). Then ya = oy or
xa - y - zaB + wB = xa - y + zaB - wB. Hence 22 = 2w = O,
or z = W’= 0 since 2 is a unit in C. Similarly since
ya = By, we obtain 2y = 2w = 0 or y = w = 0. Therefore
y e C, showing Z(A) = C.

C being local, 8(C) has the unique matrix representation

property (Theorem 3.3). Therefore there exists an integer

'33-.-. .

n and a unique central separable C-algebra D ‘which has

no idempotents except 0 and 1 such that A awmn(D).

Now D is free over C since D is projective over C
(proposition 1.17) and C is local. Therefore 4 = [A:C] =
[A:D][D:C] = n2[D:C]. There are only two possibilities:

I n=l a [D:C] = 4 =II=D = A has no non-trivial idempotents

n=2 s [D:C] 1 s A =- 9712 (C)

One checks that the element e = %'- g8 - 31 dB is a

non-trivial idempotent of A. Therefore the case n = 1 can-
not hold, and we conclude that. A ==~r77(2(C).

(2) B = (S,-1,-l) is central separable over S.
We see S=Z5 is also local with unique maximal ideal m = 525-
Hence B/mBo-(Zs/SZS,-l,-l) =- (Z/ (5),-l,-l). We have seen in
(1) that this algebra is separable over Z/(S), and there-
fore B is separable over S by Endo‘Watanabe (Theorem 1.21).
A computation identical to that in (1) will show that Z(B) = 8,

since 2 is also a unit in 8. Thus 8 is central separable

over S.

51

(3) B is an inertial S-subalgebra of A.

Since A is central separable over C, rad A = “A = (n,-1,-l)
by proposition 1.20. It is clear that S = Z5 adds with
n = 525 ® 5251. . Therefore (S,-l,—l) + (n,-1,-l) =

(C,-l,-l). That 8 is an inertial S-subalgebra now follows
from (2).

(4) m2(8) is an inertial S-subalgebra of A, but B 3?“
is pg; isomorphic to m2(s).

Now A m-m2(C) by (l), and so m2(8) is an inertial
S-subalgebra of A by proposition 2.4. Since Z c R (the

5
field of real numbers) it follows that B = (S,-1,-1) c :Ri

 

(R,-1,-l). Now (R,-l,-l) is the classical quaternion alg-
ebra which is well-known to be a division ring. Hence B
has no non-trivial idempotents, and therefore cannot be iso-
morphic to m2(8).

Remark 3.9.1: In view of Theorem 3.9, S and C are
examples of finitely generated commutative R-algebras where
S is an inertial R-subalgebra of C, but such that 8(8) 1
8(C) is not a monomorphism.

Remark 3.9.2: Define a: C * C by 0(x + Syi) = x-Syi.
Then G is an R-automorphism of C of order 2. If we let

G

<o>, then C is a connected (G,S)-algebra such that

2

IGI is a unit in 8. By the discussion following pro-
position 2.4, we see that G can be extended to A in such

a way that I = I (in this case G = I I Thus A

A C A C”
is an R-algebra which satisfies the hypotheses of Theorem

2.16. Therefore even such "well-behaved"algebras as those

52

discussed in Chapter II may possess inertial R-subalgebras

which are not isomorphic.

B IBLIOGRAPHY

10.

11.

12.

13.

BIBLIOGRAPHY

M. Auslander and 0. Goldman, The Brauer Group of a gem-
mutative Ring, Trans. Amer. Math. Soc. 91,(l960), 367-

409 O

G. Azumaya, On Maximally Central Algebras, Nagoya Math
J..; (1951), 119-150.

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