Generalized Inverses Based On Two Elements

In 2012,Drazin introduced the(b,c)-inverse of an element in a semigroup,which unifies many known generalized inverses and provides a new frame for the study of generalized inverses.This paper mainly focus on the(b,c)-inverses,(b,b)-inverses and some related generalized inverses of special elements pairs in rings and semigroups.The contents are arranged as follows:Chapter 2 is devoted to studying the structure of the set of all(b,c)-invertible elements in a semigroup.For any elements b and c of a semigroup S,denote the set of all(b,c)-invertible elements in S by S‖(b,c),and the intersection of the R-class of b and ￡-class of c by H(b,c).It is shown that if S‖(b,c)is nonempty,then the set of(b,c)-inverses of all(b,c)-invertible elements is equal to H(b,c).We first present some new equivalent conditions for H(b,c)to be a group and analyze its structure from the viewpoint of generalized inverses.Then a necessary and sufficient condition under which S‖(b,c)is a subsemigroup of S and any product of elements in S‖(b,c)satisfies the reverse order law for(b.c)-inverses is given.At last,two new operations are introduced on H(b,c)and S‖(b,c)so that these two sets become a group and a semigroup,respectively.In this case,it is shown that H(b,c)can be imbedded into S‖(b,c)and is also isomorphic to a quotient semigroup of S‖(b,c).In Chapter 3,we explore the relationship between(b,c)-inverses and(c,b)-inverses in rings and semigroups.We first provide a necessary and sufficient condition for d to be(c,b)-invertible when a is(b,c)-invertible.Then we present equivalent conditions for the existence of both all(b,c)and d‖(c,b)in terms of group invertible elements,invertible elements,(b,b)-invertible elements and(c,c)-invertible elements,which generalize the results of Cang Wu and Jianlong Chen.Furthermore,the(b,c)-invertibility is characterized by existence of some units.At last,some necessary and sufficient conditions that the inner(b,c)-inverse of a and inner(c,b)-inverse of d both exist are given,which generalize Mary’s result on inner(b,b)-inverses.In Chapter 4,we introduce and study a special class of(b,c)-inverses,i.e.the pseudo inverse along an element,which is a generalization of the {2,3}-inverse of a complex matrix with prescribed range and the pseudo core inverse.We first establish some existence criteria and formulae of the pseudo inverse along an element.Especially,it is proved that the pseudo inverse of a along b is exactly the(a,(ab)*)-inverse of a.Then we discuss the relationships between pseudo inverses along an element and some other generalized inverses including dual pseudo inverses along an element,(b,b)-inverses and Moore-Penrose inverses,which generalize the related results on pseudo core inverses.In Chapter 5,we consider the relationship between(b,b)-invertible elements and clean elements in a ring.As Nicholson proved that Drazin invertible elements are strongly clean elements,we focus on characterizing(b,b)-invertible elements by clean elements in this chapter.For any elements a,b and positive integer m,it is proved that a is(b,b)-invertible if and only if(ba)m has a clean decomposition(ba)m=e+u such that eb=0 if and only if(ab)m has a clean decomposition(ab)m=f+v such that bf=0,which improves the existence criterion of the(b,b)-inverse given by Huihui Zhu and Patricio.Then Moore-Penrose invertible elements,group invertible elements,Drazin invertible elements,(right)core invertible elements,simultaneously group and Moore-Penrose invertible elements and EP elements are characterized by clean elements,respectively.Furthermore,we give necessary and sufficient conditions for two elements both being(b,b)-invertible under a certain condition,which generalize the related results of Xiaofeng Chen and Jianlong Chen.In Chapter 6,we study the generalized inverses of elements pairs 1-ab and 1-ba,ab and ba,based on two elements a,b in a ring.Unlike the case of Drazin inverses,the pseudo core invertibility of 1-ab and 1-ba are not equivalent in general.We give necessary and sufficient conditions for 1-ba(resp.,ba)to be pseudo core invertible while 1-ab(resp.,ab)has the pseudo core inverse,meanwhile the pseudo core inverse of 1-ba(resp.,ba)is given in terms of the pseudo core inverse of 1-ab(resp.,ab).As an application,the equivalence between the pseudo core invertibility of 1-ac and 1-eae is revealed,where e is an idempotent.Inspired by the above idea,the connection between the Moore-Penrose invertibility of 1-ab and 1-ba is also considered.