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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
Matrices play an important role in both pure and applied mathematics. Matrices with entries in rings have been intensively studied and used; as well as matrices with entries in semirings, Boolean algebras, semigroups and lattices. This book focuses on a more general object: formal matrices or, alternatively, generalized matrices. What is a formal matrix? Without going into much detail, one might say that it is a matrix with elements in several different rings or bimodules.
Piotr Krylov, Askar Tuganbaev

Chapter 2. Formal Matrix Rings

Abstract
In this chapter, we define formal matrix rings of order 2 and formal matrix rings of arbitrary order n. Their main properties are considered and examples of such rings are given. We indicate the relationship between formal matrix rings, endomorphism rings of modules, and systems of orthogonal idempotents of rings. For formal matrix rings, the Jacobson radical and the prime radical are described. We find when a formal matrix ring is Artinian, Noetherian, regular, unit-regular, and of stable rank 1. In the last section, clean and k-good matrix rings are considered.
Piotr Krylov, Askar Tuganbaev

Chapter 3. Modules over Formal Matrix Rings

Abstract
We set out the basic theory of modules over formal matrix rings and consider the structure of some submodules (small and essential submodules, the socle and the radical). We pay much attention to injective, flat and projective modules over formal matrix rings of order 2.
Piotr Krylov, Askar Tuganbaev

Chapter 4. Formal Matrix Rings over a Given Ring

Abstract
If a concrete formal matrix ring of order n contains elements of some fixed ring R in all positions, then we speak about a formal matrix ring over the ring R. The properties of such rings can differ from those of the ordinary rings of all \(n\times n\) matrices over R.
Piotr Krylov, Askar Tuganbaev

Chapter 5. Grothendieck and Whitehead Groups of Formal Matrix Rings

Abstract
We develop some calculation methods for the Grothendieck group \(K_0\) and the Whitehead group \(K_1\) of formal matrix rings making use of the \(K_0\) and \(K_1\) groups of the original rings.
Piotr Krylov, Askar Tuganbaev

Backmatter

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