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Injective Modules and Injective Quotient Rings, in two parts, is
the only book of its kind to combine commutative and noncommutative
ring theory. This unique and outstanding contribution to the
mathematical literature will immediately advance the studies of
mathematicians and graduate students in the field. Written by a
leading expert in the field, Injective Modules and Injective
Quotient Rings offers readers the key concepts and methods used in
both noncommutative and commutative ring theory. Part I provides
the first non-torsion-theory proof of the Teply-Miller theorem and
the first statement and proof of the converse of the
Teply-Miller-Hansen theorem. Many applications of these theorems to
the structure of rings and modules are given, including
generalizations of theorems of Cailleau-Beck and Matlis on the
structure of -injectives and commutative rings. Part II provides an
alternative approach to the solution of Kaplansky's problem on the
classification of FGC rings. Of particular importance is the
consistent use of noncommutative ring theoretical techniques
throughout Part II to obtain theorems lying purely in the domain of
commutative ring theory. Graduate students and mathematicians in
both commutative and noncommutative ring theory will learn from the
unique approach and new general methods in ring theory contained in
Injective Modules and Injective Quotient Rings.
First published in 1982. These lectures are in two parts. Part I,
entitled injective Modules Over Levitzki Rings, studies an
injective module E and chain conditions on the set A^(E,R) of right
ideals annihilated by subsets of E. Part II is on the subject of
(F)PF, or (finitely) pseudo-Frobenius, rings [i.e., all (finitely
generated) faithful modules generate the category mod-R of all
R-modules]. (The PF rings had been introduced by Azumaya as a
generalization of quasi-Frobenius rings, but FPF includes infinite
products of Prufer domains, e.g., Z w .)
VI of Oregon lectures in 1962, Bass gave simplified proofs of a
number of "Morita Theorems", incorporating ideas of Chase and
Schanuel. One of the Morita theorems characterizes when there is an
equivalence of categories mod-A R::! mod-B for two rings A and B.
Morita's solution organizes ideas so efficiently that the classical
Wedderburn-Artin theorem is a simple consequence, and moreover, a
similarity class [AJ in the Brauer group Br(k) of Azumaya algebras
over a commutative ring k consists of all algebras B such that the
corresponding categories mod-A and mod-B consisting of k-linear
morphisms are equivalent by a k-linear functor. (For fields, Br(k)
consists of similarity classes of simple central algebras, and for
arbitrary commutative k, this is subsumed under the Azumaya [51]1
and Auslander-Goldman [60J Brauer group. ) Numerous other instances
of a wedding of ring theory and category (albeit a shot gun
wedding!) are contained in the text. Furthermore, in. my attempt to
further simplify proofs, notably to eliminate the need for tensor
products in Bass's exposition, I uncovered a vein of ideas and new
theorems lying wholely within ring theory. This constitutes much of
Chapter 4 -the Morita theorem is Theorem 4. 29-and the basis for it
is a corre spondence theorem for projective modules (Theorem 4. 7)
suggested by the Morita context. As a by-product, this provides
foundation for a rather complete theory of simple Noetherian
rings-but more about this in the introduction.
This work specifically surveys simple Noetherian rings. The authors
present theorems on the structure of simple right Noetherian rings
and, more generally, on simple rings containing a uniform right
ideal U. The text is as elementary and self-contained as
practicable, and the little background required in homological and
categorical algebra is given in a short appendix. Full definitions
are given and short, complete, elementary proofs are provided for
such key theorems as the Morita theorem, the Correspondence
theorem, the Wedderburn Artin theorem, the Goldie Lesieur Croisot
theorem, and many others. Complex mathematical machinery has been
eliminated wherever possible or its introduction into the text
delayed as long as possible. (Even tensor products are not required
until Chapter 3.)
This is the first book on the subject of FPF rings and the
systematic use of the notion of the generator of the category mod-R
of all right R-modules and its relationship to faithful modules.
This carries out the program, explicit of inherent, in the work of
G Azumaya, H. Bass, R. Dedekind, S. Endo, I. Kaplansky, K. Morita,
T. Nakayama, R. Thrall, and more recently, W. Brandal, R. Pierce,
T. Shores, R. and S. Wiegand and P. Vamos, among others. FPF rings
include quasi-Frobenius rings (and thus finite rings over fields),
pseudo-Frobenius (PF) rings (and thus injective cogenerator rings),
bounded Dedekind prime rings and the following commutative rings;
self-injective rings, Prufer rings, all rings over which every
finitely generated module decomposes into a direct sum of cyclic
modules (=FGC rings), and hence almost maximal valuation rings. Any
product (finite or infinite) of commutative or self-basic PFP rings
is FPF. A number of important classes of FPF rings are completely
characterised including semiprime Neotherian, semiperfect
Neotherian, perfect nonsingular prime, regular and self-injective
rings. Finite group rings over PF or commutative injective rings
are FPF. This work is the culmination of a decade of research and
writing by the authors and includes all known theorems on the
subject of noncommutative FPF rings. This book will be of interest
to professional mathematicians, especially those with an interest
in noncommutative ring theory and module theory.
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