Formal matrices

by Krylov, Piotr A.

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Title Statements

Full title :Formal matrices / Piotr Krylov, Askar Tuganbaev.

Summary

This monograph is a comprehensive account of formal matrices, examining homological properties of modules over formal matrix rings and summarising the interplay between Morita contexts and K theory. While various special types of formal matrix rings have been studied for a long time from several points of view and appear in various textbooks, for instance to examine equivalences of module categories and to illustrate rings with one-sided non-symmetric properties, this particular class of rings has, so far, not been treated systematically. Exploring formal matrix rings of order 2 and introducing the notion of the determinant of a formal matrix over a commutative ring, this monograph further covers the Grothendieck and Whitehead groups of rings. Graduate students and researchers interested in ring theory, module theory and operator algebras will find this book particularly valuable. Containing numerous examples, Formal Matrices is a largely self-contained and accessible introduction to the topic, assuming a solid understanding of basic algebra.

Edition Statements

Publication Date:2017.

Publication Statements

Publisher(s):Springer
Place of Publication:Cham, Switzerland
Language:eng
Publication Date:2017.

The Physical Object

Description:1 online resource

Edition Identifiers

ISBN:9783319539072, 3319539078
DOI:10.1007/978-3-319-53907-2
oclc:ocn980875046
LC Classification:QA188
Dewey Decimal:510

Symbols; 1 Introduction; 2 Formal Matrix Rings; 2.1 Construction of Formal Matrix Rings of Order 2; 2.2 Examples of Formal Matrix Rings of Order 2; 2.3 Formal Matrix Rings of Order nge2; 2.4 Some Ideals of Formal Matrix Rings; 2.5 Ring Properties; 2.6 Additive Problems; 3 Modules over Formal Matrix Rings; 3.1 Initial Properties of Modules over Formal Matrix Rings; 3.2 Small and Essential Submodules; 3.3 The Socle and the Radical; 3.4 Injective Modules and Injective Hulls; 3.5 Maximal Rings of Fractions; 3.6 Flat Modules; 3.7 Projective and Hereditary Modules and Rings.

3.8 Equivalences Between the Categories R-Mod, S-Mod, and K-Mod3.9 Hereditary Endomorphism Rings of Abelian Groups; 4 Formal Matrix Rings over a Given Ring; 4.1 Formal Matrix Rings over a Ring R; 4.2 Some Properties of Formal Matrix Rings over R; 4.3 Characterization of Multiplier Matrices; 4.4 Classification of Formal Matrix Rings; 4.5 The Isomorphism Problem; 4.6 Determinants of Formal Matrices; 4.7 Some Theorems About Formal Matrices; 5 Grothendieck and Whitehead Groups of Formal Matrix Rings; 5.1 Equivalence of Two Categories of Projective Modules; 5.2 The Group K0(A, B).

5.3 K0 Groups of Formal Matrix Rings5.…