Some New Characterizations Of EP Element And Nuclear Inverse Element

Generalized inverses originated from operator theory at first and it plays an important role in many fields.Moore-Penrose inverse(MP inverse for short)and Drazin inverse are two basic concepts in generalized inverse theory.A series of in-depth studies on this subject have been carried out and remarkable results have been obtained.In this process,many new concepts have been put forward,such as group inverse,core inverse,DMP inverse,etc.If we take into account the case that the MP inverse and the group inverse of an element are equal to each other,then the EP element is obtained.As is known to all,EP elements are always core inverses,but the converse is not true.The research results on EP elements are quite abundant and many scholars have generalized the research conclusions of EP elements to the characterizations of core inverses.This paper adopts the research method in ring theory.We study some characterizations of EP elements and core inverses in the context of associative rings with identities.Comparing their relations and differences,we can further reveal the internal relations between them.The whole paper is made up of four chapters.In Chapter 1,we briefly review the research process,findings of generalized inverse,some basic knowledge points and related theorems involved in this paper.In Chapter 2,EP elements are studied and some equivalent characterizations of EP elements are given.In Chapter 3,the equivalent characterizations between EP elements and solutions of some special equations are given.For those elements being both group invertible and MP invertible,we define a set χa and then we find certain equations such that an element of a ring is EP element if and only if these equations have solutions in χa.This is a study of EP elements from a new perspective,enriching the research contents of EP elements.In Chapter 4,the core inverse of an element is considered in a ring R with involution,the related properties are given,and some research results of the complex matrix are generalized to a ring R.Some equivalent characterizations of the core inverse of an element are proved.