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

Rings and modules which are stable under automorphisms of their injective hulls

It is proved, among other results, that a prime right nonsingular ring (in particular, a simple ring) $R$ is right self-injective if $R_R$ is invariant under automorphisms of its injective hull. This answers two questions raised by Singh and Srivastava, and Clark and Huynh. An example is given to show that this conclusion no longer holds when prime ring is replaced by semiprime ring in the above assumption. Also shown is that automorphism-invariant modules are precisely pseudo-injective modules, answering a recent question of Lee and Zhou. Furthermore, rings whose cyclic modules are automorphism-invariant are investigated.

Automorphism-invariant modules satisfy the exchange property

SURJEET SINGH AND ASHISH K SRIVASTAVADedicated to T Y Lam on his 70 th BirthdayAbstract A module is called automorphism-invariant if it is invariant underany automorphism of its injective hull In [Algebras for which every indecom-posable right module is invariant in its injective envelope, Paciﬁc J Math,vol 31, no 3 (1969), 655-658] Dickson and Fuller had shown that if R isa ﬁnite-dimensional algebra over a ﬁeld F with more than two elements thenan indecomposable automorphism-invariant right R-module must be quasi-injective In this paper we show that this result fails to hold if F is a ﬁeldwith two elements Dickson and Fuller had further shown that if Ris a ﬁnite-dimensional algebra over a ﬁeld Fwith more than two elements, then R is ofright invariant module type if and only if every indecomposable right R-moduleis automorphism-invariant We extend the result of Dickson and Fuller to anyright artinian ring A ring R is said to be of right automorphism-invarianttype (in short, RAI-type) if every ﬁnitely generated indecomposable right R-module is automorphism-invariant In this paper we completely characterizean indecomposable right artinian ring of RAI-type

Rings of Invariant Module Type and Automorphism-Invariant Modules

Dual automorphism-invariant modules

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Surjeet Singh

Papers

Rings and modules which are stable under automorphisms of their injective hulls

Rings and modules which are stable under automorphisms of their injective hulls

Rings of Invariant Module Type and Automorphism-Invariant Modules

Dual automorphism-invariant modules

Some decomposition theorems on abelian groups and their generalisations. II