Several Types Of Generalized Inverses Of Elements In Rings

In this dissertation,we mainly focus on Moore-Penrose inverses,core inverses and several types of extended Drazin inverses of elements in rings.The contents of this paper are arranged as follows.In chapter 2,several properties and characterizations on projections generated by Moore-Penrose inverses and core inverses are given in*-rings,extending the results of Hartwig and Spindelbock published on Linear Multiliear Algebra.As applications,several new characterizations of EP elements are presented in*-rings.In chapter 3,we give a positive answer to a question of Drazin published on comm.Algebra: whether an element has a unique core quasi-nilpotent decomposition implies that it is quasi-invertible in a ring.Then,we characterize the existence of the g-Drazin inverse by means of the uniqueness of a core-quasi-nilpotent decomposition defined in terms of Harte’s quasi-nilpoence in rings.In chapter 4,we introduce the concept of a generalized quasipolar element in rings and give its properties and characterizations.Then,a class of outer generalized inverses,called weakly g-Drazin inverses,is defined in rings.Moreover,it is proved that the generalized quasipolarity of a ring element coincides with the existence of its weakly g-Drazin inverse.Fianlly,several existence criteria for the weakly g-Drazin inverse are presented in rings.