Reference BookDOI
Foundations of Module and Ring Theory : A Handbook for Study and Research
TLDR
In this paper, a comprehensive introduction to module theory and the related part of ring theory, including original results as well as the most recent work, is presented and accompanied by complete proofs, where the smallest Grothendieck category containing it is denoted by o[M] and module theory is developed in this category.Abstract:
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.read more
Citations
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Journal ArticleDOI
On lifting modules
TL;DR: In this paper, it was shown that for a ring R such that every direct sum of a lifting module and a simple module is lifting, every simple R-module is small M-projective for any lifting module.
Journal ArticleDOI
The flat model structure on ()
TL;DR: In this paper, a cotorsion pair (A, B) in an abelian category C with enough A objects and enough B objects is defined, and the two pairs are related in a nice way when B is hereditary.
Journal ArticleDOI
Triangular Matrix Representations
TL;DR: In this article, the authors developed the theory of generalized triangular matrix representation in an abstract setting, which is accomplished by introducing the concept of a set of left triangulating idempotents.
Journal ArticleDOI
Rings characterized by (pre)envelopes and (pre)covers of their modules
Juan Rada,Manuel Saorín +1 more
TL;DR: In this paper, a ring characterized by (pre)envelopes and covers of their modules is described. But it is not a ring that can be used in algebraic geometry.
Journal ArticleDOI
The Torsion Theory Cogenerated by M-Small Modules
Yahya Talebi,N. Vanaja +1 more
TL;DR: In this paper, the authors consider the torsion theory cogenerated by M-small modules and show that it is cohereditary when every injective module in σ[M] is amply supplemented.
References
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Book
Rings and Categories of Modules
Frank W. Anderson,Kent R. Fuller +1 more
TL;DR: In this paper, the authors provide a self-contained account of much of the theory of rings and modules, focusing on the relationship between the one-sided ideal structure a ring may possess and the behavior of its categories of modules.
Book
An Introduction to Homological Algebra
TL;DR: An Introduction to Homological Algebra as discussed by the authors discusses the origins of algebraic topology and presents the study of homological algebra as a two-stage affair: first, one must learn the language of Ext and Tor and what it describes.
Book
A Course in Homological Algebra
Peter Hilton,Urs Stammbach +1 more
TL;DR: In this paper, the authors propose an extension of the Kunneth Theorem for Abelian groups, which is based on the notion of double complexes, and they use it to define the (co-)homology of groups.