Researches On EP Elements And Three Types Of Two-parameter Generalized Inverses And Related Issues

Generalized inverses originated from operator theory at first.When solving a system of linear equations,the matter how to deal with the situation that the coefficient matrix is singular or even a non-square matrix prompted people to con-sider the generalized inverse of a matrix,thereby generalized inverses came into being in algebra field and then achieved abundant development.The application of generalized inverses has been extended to many fields,such as mathematical s-tatistics,modern control theory,optimization theory,graph theory,network system,mathematical programming,economics,and so on.The Moore-Penrose inverse played an important role in the course of develop-ment of generalized inverses.On the one hand,its definition is simple and elegant,and also it is of great practical value.On the other hand,many other types of generalized inverses are derived from it.For instance,we have the important {1,3}-inverses amongst those which satisfy part of the conditions of the Moore-Penrose inverse,and if we take into account the case that the Moore-Penrose inverse and the group inverse of an element are equal to each other,the EP element is obtained.Some generalized inverses which are new and significative,in addition to the above-mentioned traditional generalized inverses,appeared in recent years,such as the(6,c)-inverse,which generalized the Moore-Penrose inverse,the Drazin inverse,the Chipman's weighted inverse and the Bott-Duffin inverse as its special cases,the Bott-Duffin(e,f)-inverse,which intermediates between the Bott-Duffin inverse and the(b,c)-inverse,the(e,f)-inverse,which is the basis of the inner actions of the weak Hopf algebras,and so on.A common feature to these three types of generalized inverses is that they are all defined on two elements chosen previously.For the sake of convenience,they are uniformly called,in this thesis,two-parameter generalized inverses.This thesis investigates,in the context of associative rings with identities,EP elements,three types of two-parameter generalized inverses and related issues of them.It is divided into three chapters,the first of which is introduction.Chapter two,which is divided into three sections,studies EP elements and relevant generalized inverses.{1,3}-inverses,satisfying part of the conditions of the EP elements,are explored in the first section of this chapter.We firstly present two expressions of the set of all {1,3}-inverses of an elenent.Then we discuss the necessary and sufficient conditions for an element to be {1,3}-invertible and the properties of {1,3}-invertible elements.Among other things,we generalize the concept of left*-cancellable element to that of left*-n cancellable element,and prove that element u is {1,3}-invertible if and only if u*u is regular and u is left*-2 cancellable.Next,we characterize {1,3}-invertible elements according to the existence of the solution of an equation as well as that of idempotents with particular nature.We also prove that if an {1,3}-invertible element u of ring R belongs to ZE(R),then there must exist an {1,3}-inverse of u in ZE(R).We investigate EP elements in the second section of Chapter two.Firstly,we give some equivalent conditions for an element to be EP.For those elements being both group invertible and Moore-Penrose invertible,we define a set ?a,and then we find certain equations such that an element of a ring is EP if and only if these equations have solutions in ?a-Next,we.haracterize EP elements by means of the consistency of some equations.We take advantage of the discussion about EP elements and utilize the solutions of a few equations to characterize normal elements as well as normal EP elements.It is shown that an EP element is precisely a*-strongly regular element,and therefore a ring with all its elements being EP is exactly a*-strongly regular ring.Finally we discuss the relationships between*-strongly regular rings and Abelian rings as well as*-exchange rings.The third section of Chapter two explores GEP elements and strongly EP elements.We first weaken the conditions of EP elements,obtaining the concept of GEP element.Some equivalent conditions for an element to be GEP are proposed.And it is proved that if a GEP element is either MooreoPenrose invertible or group invertible,then it is an EP element.We also study the properties of those EP elements(called strongly EP elements)which are partial isometries at the same time.We find two equations such that an element of a ring is a partial isometry or a strongly EP element if and only if these equations have solutions in ?a respectively.Chapter three studies three types of two-parameter generalized inverses and is divided into three sections.The(b,c)-inversc is discussed in the first section.We firstly characterize the(b,c)-inverse from the point of ring theory.Some necessary and sufficient conditions for an element to be(b,c)-invertible are established.Then we find a number of equations such that an element in a ring is(b,c)-invertible if and only if these equations have solutions.Finally,we propose the concept of strongly(b,c)-inversc and investigate its properties.Especially,the relationships between strongly(b,c)-inverses and EP element.s are discussed.The second section of Chapter three investigates Bott-Duffin(e,f)-inverses.Mainly,we use Bott-Duffin(e,f)-inverses to give the necessary and sufficient con-ditions for a ring to be an Abelian ring,a directly finite ring,a left min-abel ring and a strongly left min-abel ring.In the end of this section,we prove a theorem about Bott-Duffin(e,f)-inverses relating to the coradical CO of a coalgebra C in the convolution algebra HormF(C,A),which states that an element ? in the convolution algebra HomF(C,A)is a Bott-Duffin(e,f)-inverse if and only if ?0 is a Bott-Duffin(e0,f0)-inverse in HomF(C0,A).The third section of Chapter three explores(e,f)-inverses.We present certain necessary and sufficient conditions for an element in a ring to have(e,f)-inverse.and discuss the conditions which the idempotents e and must satisfy to ensure that there exist elements in the ring having(e,f)-inverses.We still use(e,f)-inverses to characterize Abelian rings,left min-abel rings and strongly left min-abel rings.