The book discusses the geometry of matrices over some rings and their applications in algebra and geometry. The first part of the book is concerned with rings and modules, matrices over a ring, affine geometry and projective geometry over a Bezout domain. Later in the book, more advanced topics, such as the geometry of rectangular matrices over a Bezout domain or a semisimple ring,the geometry of Hermilian matrices or skew-Hermitian matrices over a division ring, the geometry of symmetric matrices over a principal ideal domain and the geometry of block triangular matrices over a division ring are discussed in detail.

Preface

Notation

Chapter 1 Rings and Modules

 1.1 Rings 

 1.2 Modules and K-algebras

 1.3 Principal ideal domains

 1.4 Semisimple rings and Jacobson semisimple rings

 1.5 Rings with involution 

Chapter 2 Matrices and Modules over Ring

 2.1 Matrix over ring

 2.2 Semifir, Hermite rings and Bezout domains

 2.3 The rank of matrices

 2.4 Elementary operations and reduction of matrices over PID

 2.5 Hermitian and symmetric matrices

 2.6 Deternfinants over commutative ring

Chapter 3 Aft'me Geometry and Projective Geometry over Ring

 3.1 Subspaces of free module

 3.2 Subspaces over Bezout domain

 3.3 Affine geometry over Bezout domain

 3.4 Fundamental theorems of affine geometry over Bezout domain

 3.5 Affine geometry over division ring

 3.6 Projective geometry over rings which have IBN

Chapter 4 Geometry of Rectangular Matrices over Bezout Domains

 4.1 Geometry of rectangular matrices over division ring

 4.2 Affine geometry structures of maximal sets on matrices over Bezout

     domain

 4.3 Fundamental Theorem of geometry of rectangular matrices over

     Bezout domain

 4.4 Applications of fundamental theorem

 4.5 Projective geometry of rectangular matrices over Bezout domain

Chapter 5 Geometries of Hermitian and Skew-Hermitian Matrices over

       Division Ring

 5.1 Maximal sets of rank 1 of Hn(D)

 5.2 Adjacency preserving bijections on Hermitian matrices

 5.3 Maximal sets of rank 2 in Hn (D)

 5.4 Affine geometry structures of maximal sets of rank 2 in H2 (D)

 5.5 Geometry of 2 ~ 2 Hermitian matrices over division ring

 5.6 Fundamental theorem of geometry of nxn (n>3) Hermitian

     matrices

 5.7 Geometry of skew-Hermitian matrices over division ring

 5.8 Applications to algebra

 5.9 Projective geometry of Hermitian and symmetric matrices

Chapter 6 Geometry of Symmetric Matrices over Commutative PID

 6.1 Arithmetic distance and distance on symmetric matrices

 6.2 Maximal sets of rank one and two

 6.3 Adjacency and invertibility of determinant divisors preserving

    maps

 6.4 Fundamental theorem of geometry of symmetric matrices over

     commutative PID

Chapter 7 Geometry of Block Triangular Matrices over Division Ring

 7.1 Introduction

 7.2 Affine geometry structures on maximal sets

 7.3 Adjacency preserving bijective maps on block triangular matrices

 7.4 Fundamental theorem of geometry of block triangular matrices

 7.5 Applications of Fundamental Theorem

Chapter 8 Geometry of Matrices over Semisimple Ring

 8.1 Geometry of block diagonal matrices over some division rings

 8.2 Geometry of rectangular matrices over semisimple ring

 8.3 Geometry of block triangular matrices over simple Artinian ring

Reference

Index