Moore-Penrose Inverses And Drazin Inverses Of Elements In Rings

Moore-Penrose inve.rses and Drazin inverses are two important types of generalized inverses,which have been deeply investigated in complex matrices,Banach a.lgebras,C*-algebras and so on.In the process of studying Moore-Penrose inverses and Drazin inverses,several types of generalized inverses were introduced,such as core inverses,dual core inverses and Mary inverses.In this dissertation,we mainly consider Moore Penrose inverses,Drazin inverses,core inverses,dual core inverses and Mary inverses in semigroups and rings.In Chapter 2,we first introduce the concepts of left*-regularity and right*-regularity in a*-semigroup.Also,we prove that an element a is Moore-Penrose invertible if and only if it is left*-regular if and only if it is right*-regular,i.e.,a is Moore-Penrose invertible if and only if there exists x satisfying the equation a=aa*ax if and only if there exists y satisfying the equation a =yaa*a.In this case,a(?)=(ax)*=(ya)*.Then,we characterize the Moore-Penrose inverse of the product of three elements by one-sidded invertibilities in a*-ring.Fuurther,we consider {1,3}-inverses and {1,4}inverses of elements in a*ring.As applications,the existence eriteria and representations of(2,2,0)matrices over a ring are giver.Finally,we give the expression of the Moore Penrose inverse of 2×2 matrices over a special*-regular ring,improving the results of Hartwig and Patricio published inOper.Matrices.In Chapter 3,we study the Moore-Penrose inverse of the difference and the product of projections in a ring,and their formulae are given.We then consider the Drazin inverse of the difference and the product of idempotents.Also,necessary and sufficient conditions for the existence of the Drazin inverse of the difference and the product of idenipoterits are obtained,generalizing the results of Cvetkovic-Ilic,Deng,and those of Kolia et al.Then,the representations of the Drazin inverse of the diffence and the product of idemipotents are expressed.In Chapter 4,we consider several properties and characterizations of a centralizer in semigroups.The existence criteria of the Moore-Penrose inverse of regular elements are given in terms of centralizers and one-sided invertibility,extending classical existence criteria of Moore-Penrose inverses of Patricio and Mendes Araujo.Then,we give cen-tralizers’ applications in Drazin inverses.Moreover,the existence criteria of the Drazin inverse of the difference of two elements are given,extending the results of Deng.In Chapter 5,firstly,we give the definitions of left g-MP invertibility and right g-MP invertibility in a*-semigroup,and present some characterizations of the existence of them.Also,we prove that an element is both left g-MP invertible and right g-MP invertible if and only if it is both core invertible and dual core invertible in a*-ring.We then consider the double commutativity and the reverse order law for core inverses.Finally,we present the existence criteria of the core and dual core inverses of a regular element in terms of units in a*-ring.As applications,necessary and sufficient conditions of the existence and the formulae of the core and dual core inverses of 2×2 matrices over a ring are obtained.In Chapter 6,we introduce the concepts of one-sided Mary inverse,and give their existence criteria in a semigroup.In particular,the characterizations of one-sided Mary inverse are given by one-sided invertibility of certain elements in a ring.As applications,the existence criteria for Mary inverse of matrices over a ring are given,extending the results of Mary and Patricio published in Appl.Math.Comput.Then,the reverse order law for the Mary inverse and the existence criteria for the Mary inverse of the product of three elements are considered.Finally,we prove that the absorption law for Mary inverse holds in a ring.