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

https://curis.ku.dk/ws/files/41927598/GorensteinHomologicalDimensions.pdf

Gorenstein homological dimensions

We study classes of relative injective and projective comodules and extend well-known results about projective comodules given in [7]. The existence of covers and envelopes by these classes of comodules is also studied and used to characterize the projective dimension of a coalgebra. We also compare this homological coalgebra with the very intensively studied homological algebra of the dual algebra (see [5]).

Relative Homological Algebra

/pdf/infinite-abelian-groups-2dvcz0046a.pdf

Infinite Abelian Groups

Gorenstein injective and projective modules

Using the dual of a categorical definition of an injective envelope, injective covers can be defined. For a ringR, every leftR-module is shown to have an injective cover if and only ifR is left noetherian. Flat envelopes are defined and shown to exist for all modules over a regular local ring of dimension 2. Using injective covers, minimal injective resolvents can be defined.

Injective and flat covers, envelopes and resolvents

In this paper we give two different proofs that the flat cover conjecture is true: that is, every module has a flat cover. The two proofs are of completely different nature, and, we hope, will have different applications. The first of the two proofs (due to the third author) is essentially an application of the work of P. Eklof and J. Trlifaj (work which is more set-theoretic). The second proof (due to the first two authors) is more direct, and has a model-theoretic flavour.

Edgar E. Enochs

Papers

Relative Homological Algebra

Gorenstein injective and projective modules

Injective and flat covers, envelopes and resolvents

All modules have flat covers

8 Relative Homological Algebra and Balance