Elements of rings with equal spectral idempotents

Abstract

In this paper we define and study a generalized Drazin inverse x D for ring elements x, and give a characterization of elements a, b for which aa D = bb D . We apply our results to the study of EP elements in a ring with involution.2000 Mathematics subject classification: primary 16A32, 16A28, 15A09; secondary 46H05, 46L05.

An extension of Cline''s formula for a generalized Drazin inverse

“…Using this concept, Koliha and Patrício [6] In 2012, Wang and Chen [9] introduced an intermedium between Drazin inverse and generalized Drazin inverse. An element a ∈ R is said to be pseudo Drazin invertible if there exist b ∈ R and k ∈ N such that In 1965, Cline [1] showed that if ab is Drazin invertible then so is ba and in this case…”

On the Moore-Penrose inverse in rings with involution

“…Since ax 2 R , by [5,Theorem 5.3], we deduce that ax is *-cancellable. The rest of this proof follows as in the proof of Theorem 2.…”

Common solutions of linear equations in a ring, with applications

“…For the definition of the Moore-Penrose inverse in a ring with involution and for some of its properties, see [4]. The Moore-Penrose inverse of a ∈ R will be denoted by a † .…”