Algebras, rings and modules. Volume 2. Non-commutative algebras and rings
type: Item
instance of: scholarly article
zbMATH Open document ID: 1377.16001
author: Michiel Hazewinkel
author: Nadiya M. Gubareni
publication date: 2016-11-07T00:00:00Z
full work available at URL: https://www.taylorfrancis.com/books/9781351869874
Mathematics Subject Classification ID: 16-02
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16Pxx
zbMATH DE Number: 6648247
zbMATH Keywords: finite posets
zbMATH Keywords: primitive posets
zbMATH Keywords: quivers
zbMATH Keywords: finite dimensional algebras
zbMATH Keywords: incidence rings
MaRDI profile type: Publication
title: Algebras, rings and modules. Volume 2. Non-commutative algebras and rings
review text: Representation theory is a fundamental tool for studying groups, algebras and rings (and many other objects). This is the second volume of Algebras, rings and modules: non-commutative algebras and rings by the authors. In this volume, the authors consider representation theory for finite posets, primitive posets, quivers, finite dimensional algebras and semiperfect rings. This volume consists of eight chapters and an exhaustive list of references. Also, this book contains a glossary of ring theory.NEWLINENEWLINEChapter 1: This chapter covers the definitions and main properties of the basic main construction of rings which are incidence rings of partially ordered sets over associative rings. Also, the authors introduce a new construction of rings which are called incidence rings modulo the radical. Such rings in a more general case were introduced and studied in [\textit{M. A. Dokuchaev} et al., J. Algebra Appl. 6, No. 4, 553--586 (2007; Zbl 1162.16010)].NEWLINENEWLINEThe main aim of Chapter 2 is to present the properties and structure of various classes of rings whose lattices of submodules are distributive (or semi distributive). Such rings are called distributive (or semi distributive) and they can be considered to be a non-commutative generalization of Prüfer domains. This chapter presents properties and structure results for semiperfect semi distributive rings (SPSD-rings, in short). Also, this chapter devoted to the study of some important properties and of the structure of semihereditary SPSD-rings.NEWLINENEWLINEIn Chapter 3, an interpretation of the group \(\mathrm{Ext}_A^1(Y,X)\) (let \(A\) be a ring and let \(X, Y \in \mathrm{Mod}_rA\)) is given in terms of short exact sequences. Extensions of modules in terms of short exact sequences are also studied. They also prove the following important theorem: For any pair of \(A\) modules \(X\) and \(Y\) there is the functorial isomorphism \(\mathcal{X}: \mathrm{Ext}_A(X,Y) \to \mathrm{Ext}_A^1(Y,X)\).NEWLINENEWLINE…
reviewed by: R. Udhayakumar
description: scientific article; zbMATH DE number 6648247