(n,p)-Drazin Inverse And N Strong Generalized Inverse In Rings

The notion of generalized inverse was first introduced by Fredholm.With the devel-opment of generalized inverse theory,other types of generalized inverse have been widely studied.Drazin first proposed the definition of pseudo-inverse in rings and semigroups,now known Drazin inverse.Recently,Wang introduced the concept of strong Drazin in-verse,and Chen and Sheibani introduced the concept of Hirano inverse,which motivated Mosi(?)defined n strong Drazin inverse.Koliha introduced the concept of generalized Drazin inverse in Banach algebras,and then Koliha with Patricio extended generalized Drazin inverse to rings.Gürgün introduced generalized strong Drazin inverse,and Mosi(?)proposed n-Strong generalized Drazin inverse and pseudo n-Strong Drazin inverse.Researchers have studied invertibility and expression of a sum and a product of two elements with respect to these kind of Drazin inverse,Cline's formula and Jacobson lemma.We introduce the concept of(n,p)-Drazin inverse.Let R be a ring,a?R,n and p positive integers.If there exists x such that xax=x,ax=xa,aⁿ-a^px?R^nil,then a is said to be(n,p)-Drazin invertible,and x is the(n,p)-Drazin inverse of a.The notion of(n,p)-Drazin inverse unifies the those of Drazin inverse,strong Drazin inverse,Hirano inverse and n strong Drazin inverse.Drazin inverse is exactly(1,2)-Drazin inverse,strong Drazin inverse is exactly(1,1)-Drazin inverse,Hirano inverse is exactly(2,1)-Drazin inverse,and n strong Drazin inverse is exactly(n,1)-Drazin inverse.In Chapter 2,the concept of(n,p)-Drazin inverse is introduced,and(n,p)-Drazin inverse are characterized.It is proved that(1)if a is(n,p)-Drazin invertible,then it is Drazin invertible,and(n,p)-Drazin inverse and Drazin inverse are the same,and so(n,p)-Drazin invertible is unique if exists;(2)a is(n,p)-Drazin invertible if and only if it is|n-p+1|strong Drazin invertible,where 0 strong Drazin invertible means Drazin invertible.By means of matrix form of Peirce decomposition,(n,p)-Drazin invertibility of sums and products are discussed under certain conditions.In Chapter 3,we study generalized n strong Drazin inverses in Banach algebras.It is proved that if a?B is generalized n strong Drazin invertible,then a-aⁿ⁺¹is quasi-nilpotent.