相对于像与核的模同态广义逆研究

论文总字数：40414字

摘 要

\end{abstract}

\begin{englishabstract}

{non-commutative ring, right R-module , generalized inverse , generalized inverse with prescribed image and kernel} % 英文关键词

Generalized inverse theory is an important theoretical branch of mathematics.It plays an important role in many fields such as linear algebra,differential equations,numerical algebra,statistical inference,optimization,complex networks,Markov chains,and control theory.In 1903,Fredholm first proposed the generalized inverse of the integral operator in the study of integral equations.In 1920,Moore proposed the generalized inverse of complex matrices,but its expression is too complicated.Until 1955,Penrose proved that the MP-inverse can be represented by the solution of four matrix equations,which has aroused peoples's interest and wide attention.At this time,due to the rise and rapid development of computers,the MP-inverse theory has been deeply studied and widely used.

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In 1958,Drazin introduced pseudo-inverses in the bounding rings and semigroups,creating a new generalized inverse branch,the Drazin inverse theory.Since then,the generalized inverse theory has been continuously enrich and developed,and has introduced generalized Drazin inverse,pseudo Drazin inverse,kernel inverse,$(b,c)$-inverse,Mary inverse and so on.In 1998,Wei Yimin introduced and studied the generalized inverses of the matrix with kernel and image,such generalized inverses are common generalizations of classical generalized inverses,such as MP-inverse,Drazin inverse and group inverse.In 2005,Wang Guorong and others studied the generalized inverse of the matrix on the commutative ring with respect to the range and the kernel.In 2007,they further extended commutative ring,and studied the generalization of the matrix on the non-commutative ring with kernel and image.

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This paper studies the generalized inverse of the homomorphism of right $R$-module with image and kernel on the general ring $R$,And extend the results of Wang Guorong and others to a more general case.Specifically,there are four aspects:(1)introducing the concept of the generalized inverse of homomorphism of the right $R$-module with image and kernel,and proving its uniqueness;(2)giving the equivalent characterization of the generalized inverse of homomorphism of module with image and kernel by module direct sum and projection,and giving its exact expression;(3)if the generalized inverse of homomorphism of module with image and kernel is exist,then we can get the relationship between it and group inverse,and show the expression by group inverse;(4)by the generalized inverse of homomorphism of module with image and kernel,giving new characterizations of MP-inverse,Drazin inverse and group inverse.

% 英文摘要内容

\end{englishabstract}

\begin{Main} %正文

\chapter{基本理论与课题背景介绍}

德国数学家希尔伯特曾说过：“数学科学是一个不可分割的整体，它的生命正是在于各个部分之间的联系。”显然，代数学作为数学学科的一份子，也有着非常重要的意义。随着理论的逐渐发展与完善，代数学逐渐进入一种抽象化的阶段，那便是近世代数。近世代数，不仅仅是在现代的数学中起着非常重要的作用，其概念和方法也早已渗透于除数学以外的各个学科分支之中，其主要的研究对象是关于代数系统的代数结构，如群,环,域和模等。 %章

\section{环论的基本知识}

%节

%引用文献\cite{Korobeinikov2007}。

\par %缩进换行

通过学习近世代数(见参考文献[10])，我们已经了解了群是一种常见的含有一种代数运算的代数系统。但是，实际情况下，我们经常会遇到需要两种代数运算的时候，而环就是一种含有两种代数运算的代数系统。

%\subsection{环的定义和基本性质}

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