Algebras, rings and modules. Volume 2. Non-commutative algebras and rings (Q2832116)

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 \(\mathcal{X}: \mathrm{Ext}_A(X,Y) \to \mathrm{Ext}_A^1(Y,X)\).NEWLINENEWLINEIn Chapter 4, a number of key results on modules over semiperfect rings are given. For example, NEWLINENEWLINETheorem: Let \(A\) be a semiperfect ring. For each finitely presented non-projective right \(A\)-module \(N\) with a local endomorphism ring there exists an almost split sequence in \(\mathrm{Mod}_rA\) NEWLINE\[NEWLINE0 \to (\mathrm{Tr}N)^+ \buildrel{f}\over{\to} M \buildrel{g}\over{\to} N \to 0.NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEProposition: Let \(M\) be a finitely generated right \(A\)-module. Then there is an exact sequence \(0 \to \mathrm{Ext}^1_{A^op}(\mathrm{Tr}M, A) \to M \buildrel{\delta_M}\over{\to} M^{**} \to \mathrm{Ext}^2_{A^op}(\mathrm{Tr}M, A) \to 0\).NEWLINENEWLINEChapter 5 deals with finitely partially ordered sets (posets, in short) and their representations, which play an important role in representation theory in general. Representations of posets were first introduced and studied by \textit{L. A. Nazarova} and \textit{A. V. Roiter} [J. Sov. Math. 3, 585--606 (1972; Zbl 0336.16031); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 28, 5--31 (1972)] in 1972 in connection with problems of representations of finite dimensional algebras. This chapter also gives the proof of a criterium for primitive posets to be of finite representation type. The notion of a quiver of a finite dimensional algebra over an algebraically closed field was first introduced by \textit{P. Gabriel} in [Manuscr. Math. 6, 71--103 (1972; Zbl 0232.08001)] in connection with classification problems of finite dimensional algebras.NEWLINENEWLINEChapter 6 is concerned with quivers in the sense of Gabriel. This chapter contains 3 sections. In Section 1, the authors consider the notions of finite quivers and their representations. Section 2 is devoted to species (a generalization of quivers), valued quivers and valued graphs and their representations. In the last section, they prove some results of the representation theory of finite dimensional algebras. For example, NEWLINENEWLINETheorem. A finite dimensional algebra is either of finite representation type or there are indecomposable modules of arbitrary large finite dimension.NEWLINENEWLINEAny finite dimensional algebra of finite representation type is a semidistributive ring and the structure of Artinian semidistributive hereditary rings of finite representation type is given in Chapter 7. In the last chapter, the authors give the connection between right hereditary SPSD-rings and special kinds of \((D,\mathcal{O})\)-species and discussed the reduction of representations of \((D,\mathcal{O})\)-species to mixed matrix problems over discrete valuation rings and their common skew field of fractions. Finally, they describe right hereditary SPSD-rings of bounded representation type by reduction to representations of \((D,\mathcal{O})\)-species.NEWLINENEWLINEOverall the book is very well written. Each chapter contains many interesting results. Thus this Volume 2 of Algebras, rings and modules: non-commutative algebras and rings by the authors is accessible to a wide audience starting from students of the postgraduate. Moreover, it provides a good approach to modern representation theory for the reader. On the other hand the book serves also as an excellent reference for algebraists and scientists interested in the field of finite dimensional algebras.