環(huán)上矩陣幾何（英文版）

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