環(huán)上矩陣幾何

內(nèi)容概要

　　環(huán)與模，環(huán)上矩陣論，非交換主理想整環(huán)上的仿射幾何與射影幾何，非交換主理想整環(huán)上的長方矩陣幾何，交換主理想整環(huán)上的對稱矩陣幾何，除環(huán)上Hermitian矩陣幾何，除環(huán)上分塊三角矩陣幾何，以及矩陣幾何在代數(shù)學(xué)、幾何學(xué)、圖論等方面的應(yīng)用。

書籍目錄

PrefaceNotationChapter 1　Rings and Modules1.1　Rings1.2　Modules and K-algebras1.3　Principal ideal domains1.4　Semisimple rings and Jacobson semisimple rings1.5　Rings with involutionChapter 2　Matrices and Modules over Ring2.1　Matrix over ring2.2　Semifir， Hermite rings and Bezout domains2.3　The rank of matrices2.4　Elementary operations and reduction of matrices over PID2.5　Hermitian and symmetric matrices2.6　Deternfinants over commutative ringChapter 3　Aftme Geometry and Projective Geometry over Ring3.1　Subspaces of free module3.2　Subspaces over Bezout domain3.3　Affine geometry over Bezout domain3.4　Fundamental theorems of affine geometry over Bezout domain3.5　Affine geometry over division ring3.6　Projective geometry over rings which have IBNChapter 4　Geometry of Rectangular Matrices over Bezout Domains4.1　Geometry of rectangular matrices over division ring4.2　Affine geometry structures of maximal sets on matrices over Bezoutdomain4.3　Fundamental Theorem of geometry of rectangular matrices overBezout domain4.4　Applications of fundamental theorem4.5　Projective geometry of rectangular matrices over Bezout domainChapter 5　Geometries of Hermitian and Skew-Hermitian Matrices overDivision Ring5.1　Maximal sets of rank 1 of Hn(D)5.2　Adjacency preserving bijections on Hermitian matrices5.3　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 ring5.6　Fundamental theorem of geometry of nxn (n>3) Hermitianmatrices5.7　Geometry of skew-Hermitian matrices over division ring5.8　Applications to algebra5.9　Projective geometry of Hermitian and symmetric matricesChapter 6　Geometry of Symmetric Matrices over Commutative PID6.1　Arithmetic distance and distance on symmetric matrices6.2　Maximal sets of rank one and two6.3　Adjacency and invertibility of determinant divisors preservingmaps6.4　Fundamental theorem of geometry of symmetric matrices overcommutative PIDChapter 7　Geometry of Block Triangular Matrices over Division Ring7.1　Introduction7.2　Affine geometry structures on maximal sets7.3　Adjacency preserving bijective maps on block triangular matrices7.4　Fundamental theorem of geometry of block triangular matrices7.5　Applications of Fundamental TheoremChapter 8　Geometry of Matrices over Semisimple Ring8.1　Geometry of block diagonal matrices over some division rings8.2　Geometry of rectangular matrices over semisimple ring8.3　Geometry of block triangular matrices over simple Artinian ringReferenceIndex

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　　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.

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