Representations of the (b,c)-inverses in rings with involution

Abstract

Let R be a ring and b, c ∈ R. The concept of (b, c)-inverses was introduced by Drazin in 2012. In this paper, the existence and the expression of the (b, c)-inverse in a ring with an involution are investigated.

Outer inverses in semigroups belonging to the prescribed Green’s equivalence classes

“…After their introduction, an impressive number of papers dealt with (b, c)-inverses and inverses along an element. They have been most studied in the contexts of rings [4,5,16,36,[38][39][40]53,67,68,75,[80][81][82][83][84] and semigroups [2,14,16,[24][25][26][27][44][45][46]49], and they have been also studied in the context of Banach algebras [8,9,45], residuated semigroups and quantales [33], matrices over a field [6,15,61,70], matrices over a ring [37], tensors [59], and fuzzy matrices with entries in a complete residuated lattice [18]. Core and dual core inverses were studied in [1,43,57,69,76], in the contexts of matrices and rings, and inner and outer inverses with prescribed idempotents and ideals have been considered in [13,22,…”

One-sided (b,c)-inverses in rings

“…If such y exists, it is unique and it is called the (b, c)-inverse of a, denoted by a ⊗ . For more results on (b, c)-inverse we refer the reader to see [8,9,13,22]. In [7], M. P. Drazin also introduced the hybrid and annihilator (b, c)-inverse of a.…”

One-sided $(b, c)$-inverses in rings

“…If such y exists, it is unique and it is called the (b, c)-inverse of a, denoted by a ⊗ . For more results on (b, c)-inverse we refer the reader to see [8,9,12,19]. In [7], M. P. Drazin also introduced the hybrid and annihilator (b, c)-inverse of a.…”