Modules and algebras. Bimodule structure and group action on algebras

378 Pages·2010·1.158 MB·English

by  Wisbauer R

Modules and Algebras
...
Bimodule Structure and
Group Actions on Algebras
Robert Wisbauer
University of Du¨sseldorf
October 25, 2010
i
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Contents
Preface vii
Introduction xi
Notation xiii
1 Basic notions 1
1 Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Multiplication algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3 Identities for algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Modules over associative algebras 25
4 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5 Projectivity and generating . . . . . . . . . . . . . . . . . . . . . . . . 28
6 Injectivity and cogenerating . . . . . . . . . . . . . . . . . . . . . . . . 37
7 Regular modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
8 Lifting and semiperfect modules . . . . . . . . . . . . . . . . . . . . . . 49
3 Torsion theories and prime modules 57
9 Torsion theory in σ[M] . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
10 Singular pretorsion theory . . . . . . . . . . . . . . . . . . . . . . . . . 71
11 Polyform modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
12 Closure operations on modules in σ[M] . . . . . . . . . . . . . . . . . . 88
13 Prime modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
14 Semiprime modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4 Tensor products 123
15 Tensor product of algebras . . . . . . . . . . . . . . . . . . . . . . . . . 123
16 Modules and rings of fractions . . . . . . . . . . . . . . . . . . . . . . . 133
5 Local-global techniques 138
17 Localization at prime ideals . . . . . . . . . . . . . . . . . . . . . . . . 138
18 Pierce stalks of modules and rings . . . . . . . . . . . . . . . . . . . . . 145
19 Projectives and generators . . . . . . . . . . . . . . . . . . . . . . . . . 158
20 Relative semisimple modules . . . . . . . . . . . . . . . . . . . . . . . . 168
v
vi
6 Radicals of algebras 176
21 Radicals defined by some classes of algebras . . . . . . . . . . . . . . . 176
22 Solvable and nil ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7 Modules for algebras 190
23 The category σ[A] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
24 Generator properties of A . . . . . . . . . . . . . . . . . . . . . . . . . 198
25 Projectivity properties of A . . . . . . . . . . . . . . . . . . . . . . . . 203
26 Ideal algebras and Azumaya rings . . . . . . . . . . . . . . . . . . . . . 210
27 Cogenerator and injectivity properties of A . . . . . . . . . . . . . . . . 217
8 Separable and biregular algebras 224
28 Associative separable algebras . . . . . . . . . . . . . . . . . . . . . . . 224
29 Non-associative separable algebras . . . . . . . . . . . . . . . . . . . . . 236
30 Biregular algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
31 Algebras with local Pierce stalks . . . . . . . . . . . . . . . . . . . . . . 252
9 Localization of algebras 260
32 The central closure of semiprime rings . . . . . . . . . . . . . . . . . . 262
33 Closure operations in σ[A] . . . . . . . . . . . . . . . . . . . . . . . . . 272
34 Strongly and properly semiprime algebras . . . . . . . . . . . . . . . . 277
35 Prime and strongly prime rings . . . . . . . . . . . . . . . . . . . . . . 285
36 Localization at semiprime ideals . . . . . . . . . . . . . . . . . . . . . . 300
10 Group actions on algebras 304
37 Skew group algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
38 Associative skew group algebras . . . . . . . . . . . . . . . . . . . . . . 310
39 Generator and projectivity properties of A . . . . . . . . . . . . . . 314
A(cid:48)G
40 A as an ideal module and progenerator . . . . . . . . . . . . . . . . 323
A(cid:48)G
41 A as an M(A)(cid:48)G-module . . . . . . . . . . . . . . . . . . . . . . . . . . 329
42 The central closure of G-semiprime rings . . . . . . . . . . . . . . . . . 334
43 Examples for group actions . . . . . . . . . . . . . . . . . . . . . . . . 337
Bibliography 345
Index 361
Preface
In the theory of commutative associative algebras A, three of the most important
techniques are:
(i) the homological characterization of A in the category of A-modules,
(ii) forming the ring of quotients for prime or semiprime A and
(iii) localization at prime ideals of A.
While (i) can be successfully transferred to non-commutative associative algebras A
using left (or right) A-modules, techniques (ii) and (iii) do not allow a satifactory
extension to the non-commutative setting. However, some of the basic results in (ii)
and (iii) remain true if the algebras considered are reasonably close to commutative
rings. This closeness is attained by rings with polynomial identities (PI-rings) and
such rings have been studied in great detail by many authors. They behave like
commutative rings, e.g., in a semiprime PI-ring the non-zero ideals intersect with the
centre non-trivially. In particular, the maximal quotient ring of a prime PI-algebra is
obtained by central localization.
One of the reasons that one-sided module theory cannot imitate the commutative
case completely is that one-sided ideals are no longer kernels of ring homomorphisms.
To remedy this, one might consider the R-algebra A as an (A,A)-bimodule which is
tantamount to studying the structure of A as an A⊗ Ao-module (where Ao denotes
R
the opposite ring). Obviously the A⊗ Ao-submodules of A are precisely the ideals in
R
A, i.e., the kernels of ring homomorphisms, giving us an analogue of the commutative
case. However, in general A is neither projective nor a generator in the category
A⊗ Ao-ModofA⊗ Ao-modules andhence homological characterizations ofA, which
R R
prove to be so useful in one-sided module theory, are not possible using this technique.
Indeed, the case where A is projective in A ⊗ Ao-Mod is of special interest. Such
R
algebras are called separable R-algebras (see [58]). Central R-algebras of this type are
in fact generators in A⊗ Ao-Mod and are named Azumaya algebras. Consequently,
R
only the bimodule structure of both separable and Azumaya algebras has been the
focus of much attention.
Evenmoreseriousproblemsoccurintheattempttomakemoduletheoryaccessible
to non-associative algebras. By definition, a basic property of bimodules M over an
associative algebra A is the associativity condition:
a(mb) = (am)b, for all m ∈ M, a,b ∈ A.
This no longer makes sense for non-associative A (since A itself does not satisfy this
condition). Following a suggestion of Eilenberg (in [120]) one might try (and many
authors have) to replace this identity by suitable identities characterizing the variety
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viii Preface
of algebras under consideration. For example, a bimodule M over an alternative
algebra A should satisfy the conditions (with (−,−,−) denoting the associator):
(a,m,b) = −(m,a,b) = (b,a,m) = −(b,m,a), for all m ∈ M, a,b ∈ A.
These modules are in fact the modules over the universal enveloping algebra over A
(and thus depend on the identities of A). For this type of investigation we refer to
[166] and [41]. Looking for a more effective module theory for non-associative alge-
bras, J.M. Osborn writes in the introduction to [215]: One of the most disconserting
characteristics of the Eilenberg theory when applied to a particular variety is that the
worst-behaved bimodules occur over rings that are the best-behaved in the variety. ...
It is our feeling that the module theory used to obtain structure theory ought to be
independent of which variety the ring is thought of belonging to.
One of the (artificial) handicaps in looking for modules for non-associative alge-
bras was the desire to get a full module category, i.e., a Grothendieck category with
a finitely generated projective generator. This requirement turns out to be too re-
strictive and – for many purposes – superfluous. Indeed, it was already known from
Gabriel’s fundamental paper ([140], 1962) that most of the localization techniques are
available in any Grothendieck category (since it has enough injectives). Consequently,
we may meet our localization needs by finding a Grothendieck category related to A
whichdoesnotnecessarilyhaveaprojectivegenerator. Ontheotherhand,forahomo-
logical characterization of an algebra A it is essential that the objects in the category
used are closely related to A. Both objectives are met in the following construction.
For any R-algebra A we consider the multiplication algebra M(A), i.e., the R-
subalgebra of End (A) generated by left and right multiplications by elements of
R
A and the identity map of A. Then A is a left module over the unital associative
algebra M(A) and we denote by σ[A] the smallest full Grothendieck subcategory of
M(A)-Mod containing A. The objects of σ[A] are just the M(A)-modules which are
submodules of A-generated modules. This category is close enough to A to reflect
(internal) properties of A (see (i)) and rich enough for the constructions necessary
for (ii) and (iii). Moreover, the construction is independent of the variety to which
A belongs. (This will be studied in detail in the second part of this monograph.) In
particular it extends the study of bimodules over associative algebras by M. Artin [57]
and Delale [115] to arbitrary algebras.
It is easy to see that for A finitely generated as an R-module, σ[A] = M(A)-Mod.
Moreover, if A is associative and commutative with unit then σ[A] = A-Mod. Hence
σ[A] generalizes the module theory over associative commutative rings. To measure
how close an algebra A is to an associative commutative algebra, we consider its
behaviour as an M(A)-module instead of the aforementioned polynomial identities.
For example, we may ask if σ[A] = M(A)-Mod, or if Hom (A,U) (cid:54)= 0 for non-
M(A)
zero ideals U ⊂ A. The conditions which determine when these occur do not depend
Preface ix
on associativity. For associative (and some non-associative) prime algebras they are
conveniently equivalent to A satisfying a polynomial identity.
The use of the multiplication algebra to investigate (bimodule) properties of any
algebra is by no means new. A footnote in Albert [42] says: The idea of studying these
relations was suggested to both Jacobson and the author (Albert) by the lectures of H.
Weyl on Lie Algebras which were given in Fine Hall in 1933. One of the early results
in this context is the observation that a finite dimensional algebra A over a field is a
direct sum of simple algebras if and only if the same is true for M(A) (see Jacobson
[165], 1937; Albert [43], 1942). Later, in Mu¨ller [208] separable algebras A over a ring
R were defined by the R-separability of M(A), provided A is finitely generated and
projective as an R-module. A wider and more effective application of module theory
to the structure theory of algebras was made possible through the introduction of
the category σ[A]. In particular, in this case A need not be finitely generated as an
R-module (which would imply σ[A] = M(A)-Mod).
The category σ[A] is a special case of the following more general situation. Let
M be any module over any associative ring A and denote by σ[M] the smallest full
subcategory of A-Mod which is a Grothendieck category. Its objects are just the
submodules of M-generated modules. Clearly, for M = A we have σ[M] = A-Mod.
Many results and constructions in A-Mod can be transferred to σ[M] and this is
done in [11, 40]. In the first part (Chaps. 2, 3) of this monograph we will recall
some of these results and introduce new ones which are of particular interest for the
applications we have in mind. For example, we consider generating and projectivity
properties of M in σ[M] and also special torsion theories in σ[M].
The first application is – as mentioned above – the investigation of any algebra A
as a module over its multiplication M(A).
For another application we will use our setting to study the action of a group G
on any algebra A. In the case A is unital and associative we may consider A as a left
module over the skew group algebra A(cid:48)G. The endomorphism ring of this module is
the fixed ring AG and applying our techniques we obtain relations between properties
of A and AG.
A(cid:48)G
For arbitrary A with unit we observe that the action of G on A can be extended
to an action on M(A). This allows us to consider A as a module over the skew group
algebraM(A)(cid:48)G. Theendomorphismringofthismoduleconsistsofthefixedelements
of the centroid.
.
x
Introduction
As pointed out in the preface, the structure of algebras can be studied by methods
of associative module theory. The purpose of this monograph is to give an up-to-
date account of this theory. We begin in Chapter 1 by presenting those topics of the
associative theory which are of relevance to the bimodule structure of algebras.
In Chapter 2 we collect results on modules M over associative algebras A and the
related category σ[M], a full subcategory of the category of all left A-modules whose
objects are submodules of M-generated modules. In addition to information taken
from the monographs [40] and [11] new notions are introduced which will be helpful
later on. This leads on to Chapter 3 where we outline the localization theory in σ[M].
IfAisanassociativealgebraoveranassociative, commutativeringR, everyleftA-
module M is also an R-module and there is an interplay between the properties of the
A-module M and the R-module M. In particular the tensor product of A-modules
can be formed over R. This facilitates the study of localization of A and M with
respect to multiplicative subsets of R. These techniques are considered in Chapter 4.
Then they are applied in Chapter 5 to obtain local-global characterizations of various
module properties.
Next, inwhatfollows, AwillbeanotnecessarilyassociativeR-algebra. InChapter
6 some radicals for such algebras are considered. In Chapters 7 to 9 the module
theory presented earlier will be applied to the subcategory σ[A] of M(A)-Mod, where
M(A) denotes the multiplication algebra of A. Generating and projectivity properties
of A as an M(A)-module are considered and Azumaya rings are characterized as
projective generators in σ[A], whereas Azumaya algebras are (projective) generators
in M(A)-Mod. The effectiveness of localization in σ[A] for semiprime algebras A is
based on the observation that such algebras are non-A-singular as M(A)-modules.
This yields in particular an interpretation of Martindale’s central closure of A as the
injective envelope of A in σ[A].
In Chapter 10 the module theoretic results are used to study the action of groups
on any algebra A. If A is associative and unital we consider A as an algebra over the
skew group ring A(cid:48)G. In particular we ask when A is a self-generator or when it is
A(cid:48)G
self-projective and what are the properties of the fixed ring AG (which is isomorphic
to the endomorphism ring of A). This generalizes the case when A is a projective
A(cid:48)G
generator in A(cid:48)G-Mod, a property which has attracted interest in connection with
Galois theory for rings.
For arbitrary A with unit we use the fact that the action of G on A induces an
action on the multiplication algebra M(A). Hence we have A as module over the skew
group algebra M(A)(cid:48)G whose endomorphism ring consists of the fixed elements of the
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xii Introduction
centre. As a counterpart to the central closure of prime rings, we obtain a central
quotient ring for G-semiprime algebras.
Throughout the monograph R will denote a commutative associative ring with
unit. A will be any R-algebra which in some chapters is assumed to be associative
or to have a unit. For the general properties of σ[A] associativity of A is of no
importance. However, any (polynomial) identity on A may imply special properties
of A as an M(A)-module.
For the readers convenience, each section begins with a listing of its paragraph
titles. Most of the sections are ended by exercises which are intended to point out
further relationships and to draw attention to related results in the literature.
I wish to express my sincere thanks to all the colleagues and friends who helped
to write this book. In particular I want to mention Toma Albu, John Clark, Maria
Jos´e Arroyo Paniagua, Jos´e R´ıos Montes and my students for their interest and the
careful reading of the manuscript. Several interesting results around semiprime rings
were obtained in cooperation with Kostja Beidar and Miguel Ferrero. Moreover I am
most indebted to Bernd Wilke for the present form of Chapter 10 and to Vladislav
Kharchenko for many helpful comments.
Du¨sseldorf, January 1996
Robert Wisbauer