This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

This volume provides a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work. It is a useful and stimulating study for those new to the subject as well as for researchers and serves as a reference volume. Starting form a basic understanding of linear algebra, the theory is presented and accompanied by complete proofs. For a module M, the smallest Grothendieck category containing it is denoted by o[M] and module theory is developed in this category. Developing the techniques in o[M] is no more complicated than in full module categories and the higher generality yields significant advantages: for example, module theory may be developed for rings without units and also for non-associative rings. Numerous exercises are included in this volume to give further insight into the topics covered and to draw attention to related results in the literature.

Foundations of Module and Ring Theory : A Handbook for Study and Research

An introduction to noncommutative noetherian rings

Let T be a set of elements in a ring A. The set T is right permutable if for any a ∈ A and t ∈ T, there exist b ∈ A, u ∈ T such that au = tb. A multiplicative set in a ring A is any subset T of A such that 1 ∈ T,0 ∉ T and T is closed under multiplication. A completely prime ideal in a ring A is any proper ideal B such that A\B is a multiplicative set (i.e. A/Bis a domain). A minimal prime ideal (resp. minimal completely prime ideal) in a ring A is any prime (resp. completely prime) ideal P such that P contains no properly any other prime ideal (resp. completely prime ideal) of A. Let I be any proper ideal of a ring A. The set of all elements a ∈ A such that a + I is a regular element of A/I is denoted by c(I). In particular, c(0) is the set of all regular elements of A.

Rings of quotients

A1 Regular and Cartesian Closed Categories A2 Toposes - Basic Theory A3 Allegories A4 Geometric Morphisms - Basic Theory B1 Fibrations and Indexed Categories B2 Internal and Locally Internal Categories B3 Toposes over a base B4 BTop/S as a 2-Category

REVIEWS-Sketches of an elephant: A topos theory compendium

1 Basic Concepts.- 1.1 Semisimple rings and modules.- 1.2 Local and semilocal rings.- 1.3 Serial rings and modules.- 1.4 Pure exact sequences.- 1.5 Finitely definable subgroups and pure-injective modules.- 1.6 The category (RFP, Ab).- 1.7 ?-pure-injective modules.- 1.8 Notes on Chapter 1.- 2 The Krull-Schmidt-Remak-Azumaya Theorem.- 2.1 The exchange property.- 2.2 Indecomposable modules with the exchange property.- 2.3 Isomorphic refinements of finite direct sum decompositions.- 2.4 The Krull-Schmidt-Remak-Azumaya Theorem.- 2.5 Applications.- 2.6 Goldie dimension of a modular lattice.- 2.7 Goldie dimension of a module.- 2.8 Dual Goldie dimension of a module.- 2.9 ?-small modules and ?-closed classes.- 2.10 Direct sums of ?-small modules.- 2.11 The Loewy series.- 2.12 Artinian right modules over commutative or right noetherian rings.- 2.13 Notes on Chapter 2.- 3 Semiperfect Rings.- 3.1 Projective covers and lifting idempotents.- 3.2 Semiperfect rings.- 3.3 Modules over semiperfect rings.- 3.4 Finitely presented and Fitting modules.- 3.5 Finitely presented modules over serial rings.- 3.6 Notes on Chapter 3.- 4 Semilocal Rings.- 4.1 The Camps-Dicks Theorem.- 4.2 Modules with semilocal endomorphism ring.- 4.3 Examples.- 4.4 Notes on Chapter 4.- 5 Serial Rings.- 5.1 Chain rings and right chain rings.- 5.2 Modules over artinian serial rings.- 5.3 Nonsingular and semihereditary serial rings.- 5.4 Noetherian serial rings.- 5.5 Notes on Chapter 5.- 6 Quotient Rings.- 6.1 Quotient rings of arbitrary rings.- 6.2 Nil subrings of right Goldie rings.- 6.3 Reduced rank.- 6.4 Localization in chain rings.- 6.5 Localizable systems in a serial ring.- 6.6 An example.- 6.7 Prime ideals in serial rings.- 6.8 Goldie semiprime ideals.- 6.9 Diagonalization of matrices.- 6.10 Ore sets in serial rings.- 6.11 Goldie semiprime ideals and maximal Ore sets.- 6.12 Classical quotient ring of a serial ring.- 6.13 Notes on Chapter 6.- 7 Krull Dimension and Serial Rings.- 7.1 Deviation of a poset.- 7.2 Krull dimension of arbitrary modules and rings.- 7.3 Nil subrings of rings with right Krull dimension.- 7.4 Transfinite powers of the Jacobson radical.- 7.5 Structure of serial rings of finite Krull dimension.- 7.6 Notes on Chapter 7.- 8 Krull-Schmidt Fails for Finitely Generated Modules and Artinian Modules.- 8.1 Krull-Schmidt fails for finitely generated modules.- 8.2 Krull-Schmidt fails for artinian modules.- 8.3 Notes on Chapter 8.- 9 Biuniform Modules.- 9.1 First properties of biuniform modules.- 9.2 Some technical lemmas.- 9.3 A sufficient condition.- 9.4 Weak Krull-Schmidt Theorem for biuniform modules.- 9.5 Krull-Schmidt holds for finitely presented modules over chain rings.- 9.6 Krull-Schmidt fails for finitely presented modules over serial rings.- 9.7 Further examples of biuniform modules of type 1.- 9.8 Quasi-small uniserial modules.- 9.9 A necessary condition for families of uniserial modules.- 9.10 Notes on Chapter 9.- 10 ?-pure-injective Modules and Artinian Modules.- 10.1 Rings with a faithful ?-pure-injective module.- 10.2 Rings isomorphic to endomorphism rings of artinian modules.- 10.3 Distributive modules.- 10.4 ?-pure-injective modules over chain rings.- 10.5 Homogeneous ?-pure-injective modules.- 10.6 Krull dimension and ?-pure-injective modules.- 10.7 Serial rings that are endomorphism rings of artinian modules.- 10.8 Localizable systems and ?-pure-injective modules over serial rings.- 10.9 Notes on Chapter 10.- 11 Open Problems.

Module Theory: Endomorphism rings and direct sum decompositions in some classes of modules

We answer a question posed by Warfield in 1975: the KrullSchmidt Theorem does not hold for serial modules, as we show via an example. Nevertheless we prove a weak form of the Krull-Schmidt Theorem for serial modules (Theorem 1.9). And we show that the Grothendieck group of the class of serial modules of finite Goldie dimension over a fixed ring R is a free abelian group. In 1975 R. B. Warfield published a very interesting paper [8], in which he described the structure of serial rings and proved that every finitely presented module over a serial ring is a direct sum of uniserial modules. On page 189 of that paper, talking of the problems that remained open, he said that “ . . . perhaps the outstanding open problem is the uniqueness question for decompositions of a finitely presented module into uniserial summands (proved in the commutative case and in one noncommutative case by Kaplansky [5]).” We solve Warfield’s problem completely: Krull-Schmidt fails for serial modules. The two main ideas in this paper are the epigeny class and monogeny class of a module. We say that modules U and V are in the same monogeny class, and we write [U ]m = [V ]m, if there exist a module monomorphism U → V and a module monomorphism V → U . In the same spirit, we say that U and V are in the same epigeny class, and write [U ]e = [V ]e, if there exist a module epimorphism U → V and a module epimorphism V → U . The significance of these definitions is that uniserial modules U, V are isomorphic if and only if [U ]m = [V ]m and [U ]e = [V ]e (Proposition 1.6). Our technical starting point is that the endomorphism ring of a uniserial module has at most two maximal ideals, and modulo those ideals it becomes a division ring (Theorem 1.2). We show (Theorem 1.9) that if U1, . . . , Un, V1, . . . , Vt are non-zero uniserial modules, then U1 ⊕ · · · ⊕ Un ∼= V1 ⊕ · · · ⊕ Vt if and only if n = t and there are two permutations σ, τ of {1, 2, . . . , n} such that [Uσ(i)]m = [Vi]m and [Uτ(i)]e = [Vi]e for every i = 1, 2, . . . , n. And we show that for every n ≥ 2 there exist 2n pairwise non-isomorphic finitely presented uniserial modules U1, U2, . . . , Un, V1, V2, . . . , Vn over a suitable serial ring such that U1 ⊕ U2 ⊕ · · · ⊕ Un ∼= V1 ⊕ V2 ⊕ · · · ⊕ Vn (Example 2.2). The weakened form of the Krull-Schmidt Theorem that serial modules satisfy (Theorem 1.9) is sufficient to allow us to compute the Grothendieck group of the class of serial modules of finite Goldie dimension over a fixed ring R. As is well Received by the editors August 4, 1995. 1991 Mathematics Subject Classification. Primary 16D70, 16S50, 16P60. Partially supported by Ministero dell’Università e della Ricerca Scientifica e Tecnologica (Fondi 40% e 60%), Italy. This author is a member of GNSAGA of CNR. c ©1996 American Mathematical Society

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Alberto Facchini

Papers

Module Theory: Endomorphism rings and direct sum decompositions in some classes of modules

Krull-Schmidt fails for serial modules

Direct sum decompositions of modules, semilocal endomorphism rings, and Krull monoids☆

Rings, Modules, Algebras, and Abelian Groups

Rings of pure global dimension zero and Mittag-Leffler modules