Bicommuting properties of generalized inverses

Abstract

Let R be any associative ring and let . By making use of the bicommutant comm, Koliha and Patrício [Elements of rings with equal spectral idempotents. J Aust Math Soc. 2002;72:137–152], Wang and Chen [Pseudo Drazin inverses in associative rings and Banach algebras. Linear Algebra Appl. 2012;437:1332–1345], and Drazin [Generalized inverses: uniqueness proofs and three new classes. Linear Algebra Appl. 2014;449:402–416] have found three ways to define a unique idempotent p associated with a for increasingly general a. In this article it is shown that one can associate with a unique ‘left’ and ‘right’ idempotents p and q under much weaker conditions on a. For any semigroup S and any given , in 2012 [Drazin MP. A class of outer generalized inverses. Linear Algebra Appl. 2012;436:1909–1923] the author defined as being a (b, c)-inverse of a if , and , which hypotheses are satisfied, for suitable b and c, by essentially every known outer generalized inverse y. Here it is shown that if b and c are equivalent under J.A. Green’s equivalence relation (or, in particular, if , as holds for the Moore–Penrose inverse , for the pseudo-inverse and for the Bott-Duffin inverse), then and . This leads to weaker alternative conditions on a, b, c still sufficient for the existence of unique corresponding left and right idempotents p, q. Beyond comm, etc., a more general version is established which replaces ordinary commutativity by an intertwining property.