The Existence And Representations Of Drazin Inverse In Rings

Abstract:Drazin inverse is a kind of very important generalized inverse, which is widely used in many significant fields. Since Drazin inverse is introduced, a number of scholars study around it on the complex matrices, Banach algebras, rings and semigroups. Although abundant research achievements have been made, but there are still a lot of problems remain to be further discussed. This paper mainly discusses the group inverse of 2 x 2 partitioned matrices in rings and the existence and representations of Drazin inverse of the sum and product of two elements in rings, and it consists of two parts.The first part using the methods and skills of the ring theory, first of all, discusses the existence and representations of group inverse of (2,2,0) partitioned matrices in rings. The main result as follows:let M=(AB CO), where R(A) (?) R(B), Rr(A) (?) Rr(C), then M is group invertible if and only if R(B)= R(BC), R(C)= R(CB), Rr(B)= Rr(CB), Rr(C)= Rr(BC). This generalized the relevant conclusion of X.F. Song etc. Secondly, we discuss the existence and representations of group inverse of 2 x 2 partitioned matrices in rings. The main result as follows:let M= (ABCD), if there are X, Y such that CXA= C, AYB= B, and S= D-CXB is group invertible, then M is group invertible if and only if F= A2+BSπC is regular, and R(F)= R(A), RT(F)= RT(A). This generalized the result of C.G. Cao etc.The second part mainly discusses the existence and the representations of Drazin inverse of the sum and product of two elements in rings. First of all, let p, q are two idempotents in an algebra A over any field F. On the one hand, the existence and representations of Drazin inverse of multiple combination of p, q are obtained, when (pq)2= pq and pqp=λp (λ∈ F) be satisfied respectively. On the other hand, the existence and representations of Drazin inverse of linear combination of p, q are obtained, when (pq)m= (pq)m+1, (pq)m≠(pq)m-1 (m≥1) be satisfied. These results generalized the results on complex matrices and Banach algebras got by C.Y. Deng and Y.F. Shi etc to an algebra over any field. In addition, the group invertibility of a+b is investigated when a, b are group invertible and satisfy baa#= abb#aa#= baa#bb# in a Dedekind-finite ring. This result generalized related results of X. Zhu and X.J. Liu etc.