Two Classes Studies Of Generalized Inverses In Rings

Research on generalized inverses in rings plays the important part in the ring theory.The theory and method of the generalized inverses are widely applied in many field of mathematics.Some important generalized inverses sprang up in the study()f goneralized inverses.For instance,Drazin inverses.An element a in a ring R is said to be(generalized)Drazin invertible if there exists an element x∈R such that xax=x,xa=ax and a-a2x is(quasinilpotent)nilpotent,x is called the(generalized)Drazin inverse of a.(generalized)Drazin inverses are closely related to(quasipolar)strongly π-regular elements.In this thesis,we study generalized strongly Drazin inverses,define and study J-Drazin inverses.First,the background,research status,symbols are presented in brief and main work and basic concepts.Next,we define the J-Drazin inverses.Basic properties of J-Drazin inverses are investigated.We prove that a∈R is J-Drazin invertible if and only if a∈R is J-quasipolar.Moreover,relations among J-Drazin inverses.pseudo Drazin inverses and generalized Drazin inverses are discussed.Cline’s formula and the Jacobson s lemma for the J-Drazin inverses are given in rings.We also study properties of J-Drazin inverses in Banach algebras.Finally,we study properties of generalized strongly Drazin inverses in Banach algebras.We prove that a∈R is generalized strongly Drazin invertible if and only if a∈ R is strongly quasi-nil clean.Examples of generalized strongly Drazin invortible are given.Moreover,let a.b ∈A be generalized strongly Drazin invertible and ab=ba.Then a+b is generalized strongly Drazin invertible if and only if 1+aqb is generalized strongly Drazin invertible,and aq is generalized strongly Drazin inverse of a.