This book is an introduction to the theory of associative rings and their modules, designed primarily for graduate students. The standard topics on the structure of rings are covered, with a particular emphasis on the concept of the complete ring of quotients. A survey of the fundamental concepts of algebras in the first chapter helps to make the treatment self-contained. The topics covered include selected results on Boolean and other commutative rings, the classical structure theory of associative rings, injective modules, and rings of quotients. The final chapter provides an introduction to homological algebra. Besides three appendices on further results, there is a six-page section of historical comments. Table of Contents: Fundamental Concepts of Algebra: 1.1 Rings and related algebraic systems; 1.2 Subrings, homomorphisms, ideals; 1.3 Modules, direct products, and direct sums; 1.4 Classical isomorphism theorems. Selected Topics on Commutative Rings: 2.1 Prime ideals in commutative rings; 2.2 Prime ideals in special commutative rings; 2.3 The complete ring of quotients of a commutative ring; 2.4 Rings of quotients of commutative semiprime rings; 2.5 Prime ideal spaces. Classical Theory of Associative Rings: 3.1 Primitive rings; 3.2 Radicals; 3.3 Completely reducible modules; 3.4 Completely reducible rings; 3.5 Artinian and Noetherian rings; 3.6 On lifting idempotents; 3.7 Local and semiperfect rings. Injectivity and Related Concepts: 4.1 Projective modules; 4.2 Injective modules; 4.3 The complete ring of quotients; 4.4 Rings of endomorphisms of injective modules; 4.5 Regular rings of quotients; 4.6 Classical rings of quotients; 4.7 The Faith-Utumi theorem. Introduction to Homological Algebra: 5.1 Tensor products of modules; 5.2 Hom and $\otimes$ as functors; 5.3 Exact sequences; 5.4 Flat modules; 5.5 Torsion and extension products. Appendixes; Comments; Bibliography; Index. Review from Zentralblatt Math: Due to their clarity and intelligible presentation, these lectures on rings and modules are a particularly successful introduction to the surrounding circle of ideas. Review from American Mathematical Monthly: An introduction to associative rings and modules which requires of the reader only the mathematical maturity which one would attain in a first-year graduate algebra [course]...in order to make the contents of the book as accessible as possible, the author develops all the fundamentals he will need. In addition to covering the basic topics...the author covers some topics not so readily available to the nonspecialist...the chapters are written to be as independent as possible...[which will be appreciated by] students making their first acquaintance with the subject...one of the most successful features of the book is that it can be read by graduate students with little or no help from a specialist. (CHEL/283.H)

Lectures on Rings and Modules

0. Throughout this paper we assume that every ring has a unit element. A module is called injective if it is a direct summand of every extension module. A ring is said to be left self injective if it is iniective as the left module over itself. The main results we shall show in the present paper are the following: Let S be a left self injective ring. Then S/N(S) is also left self injective, where N(S) denotes the Jacobson radical of S. Any system of orthogonal idempotents of S/N(S) can be lifted to a system of orthogonal idempotents of S. This theorem about orthogonal idempotents can be proved under a somewhat weaker assumption than the left self injectivity of S. In fact, it is enough to suppose that S is a ring satisfying the following two conditions: 0.1. CONDITION. For any left ideal A there is an idempotent e such that Se is an essential extension of A.

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On continuous rings and self injective rings

On quotient rings

Quasi-frobenius (= QF) rings have many interesting characterizations. One such, due to Ikeda [17] is that these rings are right (left) artinian and right (left) self-injective. Thus, if R is QF, then R is right (left) noetherian, so each direct sum of injective right R-modules is injective: in particular, each free, hence, each projective, R-module is injective. One object of this paper is to report that this property characterizes QF-rings:

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Rings with ascending condition on annihilators

This paper is an introduction: in particular, to algebras of relations of various ranks, and in general, to the part of algebraic logic algebraizing quantifier logics. The paper has a survey character, too. The most frequently used algebras like cylindric-, relation-, polyadic-, and quasi-polyadic algebras are carefully introduced and intuitively explained for the nonspecialist. Their variants, connections with logic, abstract model theory, and further algebraic logics are also reviewed. Efforts were made to make the review part relatively comprehensive. In some directions we tried to give an overview of the most recent results and research trends, too.

Algebraization of quantifier logics, an introductory overview

Quasi-Injective Modules and Irreducible Rings

In a recent paper,' K. Asano gave a new proof of the theorem that a domain of integrity has a right quotient ring if and only if every pair of nonzero elements has a common nonzero right multiple. His method of proof is used in the present work to extend the centralizer of a ring over a module to a system of semi-endomorphisms of the module. From this extension, necessary and sufficient conditions that a ring have a right quotient regular ring are derived. Consider a given ring R, and a given nonzero right R-module M. Denote by 9) the set of all submodules of M, and by 9X* the set of all submodules N of M having the property that NQN'00 for all nonzero N'EC9). Since MECP*, 9* is not void. It is easily seen that if N and N' are in 9*, then N+N' and N \NY' are also in 1*. Thus {*; C, (, + } is a sublattice of the lattice {I Y; C, C, + }. An R-homomorphism of N into M, N any element of 9), is called a semi-endomorphism of M. Thus, thinking of the semi-endomorphism a as a left operator on N, we have av(x+y) = ax+ ay and a(xa) = (ax)a for all x, yCN, aCR. For convenience, the module N on which a is defined is denoted by Ma. The set of all semi-endomorphisms of M is labeled with 9. Contained in 9 is the usual centralizer of R over M consisting of all a E9 for which Ma =M. A partial ordering < is defined in 9 as follows: a <? if and only if MaCM# and ax= 3x for all xCMa. The notation a <f3 is used in case a?_<3 and Ma,,M. In case 2 is a linearly ordered subset of 2, and M'= UMa,, atES, the mapping y of M' into M defined by

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The extended centralizer of a ring over a module

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Quotient rings of rings with zero singular ideal.

This paper is a continuation of the study of prime rings started in [2]. We recall that a prime ring is a ring having its zero ideal as a prime ideal. A right (left) ideal I of a prime ring R is called prime if abCI implies that acI (bCI), a and b right (left) ideals of R with b5O (aXO). We denote by r (i$1) the set of all prime right (left) ideals of R. For any subset A of R, Ar (Al) denotes the right (left) annihilator of A; Ar (A1) is a right (left) annihilator ideal of R. The set of all right (left) annihilator ideals of R is denoted by W, (1). For the prime rii'gs R studied in [2 ], it was assumed that there existed a mapping I-* of the set of all right (left) ideals of R onto a subset T? (3) of 1r ($3) having the following seven properties:

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R. E. Johnson

Papers

Quasi-Injective Modules and Irreducible Rings

The extended centralizer of a ring over a module

Quotient rings of rings with zero singular ideal.

Representations of prime rings

Semi-prime rings