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

On endomorphism rings of quasiprojective modules

Throughout R is a ring with identity and all modules are unital right /^-modules. The category of all such modules is denoted by mod-R. A module M is called semi-artinian (or a Loewy module) if every non-zero homomorphic image of M contains a simple submodule or, equivalently, if every non-zero homomorphic image of M has essential socle. Dually, a module M is semi-noetherian if every non-zero submodule contains a maximal submodule. It is well known (see, for example, [1, §11]) that a module M has finite length if and only if it is semi-artinian and Noetherian or, equivalently, it is seminoetherian and Artinian. The main result of this note shows that we can often get finite length by just assuming that M is semi-artinian, semi-noetherian and has finitely generated socle. As a consequence of this we can obtain a recent characterization of quasi-Frobenius rings in [9]. The ring R is called right semi-artinian (respectively right semi-noetherian) if the right /^-module R is semi-artinian (semi-noetherian). It is well known (see, for example, [10, Proposition 22.32]) that if R is right semi-artinian then each non-zero M in mod-K is also semi-artinian. Chapter 22 of [10] contains further information on semi-artinian rings and rings for which every module is semi-noetherian, therein called socular and firings respectively. The relationship between these two classes of rings is considered in [24]. In particular [24, Theoreme 3.1] shows that any commutative semi-artinian ring is semi-noetherian. For any module M, E(M) will denote its injective hull. The socle of M will be denoted by Soc(M). The second socle Soc2(M) is the submodule of M containing Soc(M) such that Soc2(M)/Soc(M) = Soc(M/Soc(M)).

On semi-artinian modules and injectivity conditions

Properties of endomorphism rings

All rings are assumed to be associative and with nonzero identity element. A module M is said to be distributive if its submodule lattice Lat(M) is distributive. A module M is said to be uniserial if any two of its submodules are comparable with respect to inclusion (i.e., the lattice Lat(M) is a chain). Expressions such as “a distributive ring” mean that the corresponding right and left conditions hold. In this work, we present some results on distributive rings of [9,10], and [11]. The main results are Theorems 1.1, 1.2, 2.1, 2.2, 2.3, 3.1, and 3.2 formulated in Sections 1, 2, and 3. For a ring A, we denote by J(A), N (A), Nr(A), N(A), Z(AA), Z(AA), Qmax(A), and maxQ(A) the Jacobson radical, the set of all left divisors of zero, the set of all right divisors of zero, the set of all left or right divisors of zero, the right singular ideal, the left singular ideal, the maximal right ring of fractions, and the maximal left ring of fractions, respectively. It is clear that N(A) = Nr(A) ⋃ N (A). The set A \N(A) is denoted by T (A), and the elements of the set T (A) are called regular elements (in A).

A.K. Gupta

Papers

Modules over endomorphism rings