The (b, c)-inverse in semigroups and rings with involution

TL;DR

This paper explores properties of the (b, c)-inverse in semigroups and rings with involution, extending existing results and characterizations of EP elements and invertibility conditions.

Contribution

It generalizes recent results on inverse along an element and extends Moore-Penrose inverse properties to (b, c)-inverses in *-monoids.

Findings

A left (b, c)-invertible element that is also left (c, b)-invertible is (b, c)-invertible and (c, b)-invertible.

Conditions under which an element is EP, one-sided core invertible, or group invertible are characterized.

Extensions of Moore-Penrose inverse results to (b, c)-inverses are established.

Abstract

In this paper, we first prove that if a is both left (b, c)-invertible and left (c, b)-invertible, then a is both (b, c)-invertible and (c, b)-invertible in a *-monoid, which generalized the recent result about the inverse along an element by Wang and Mosic, under the conditions (ab)* = ab and (ac)* = ac. In addition, we consider that ba is (c, b)- invertible, and at the same time ca is (b, c)-invertible under the same conditions, which extend the related results about Moore-Penrose inverses by Chen et al. to (b, c)-inverses. As applications, we obtain that under condition (a2)* = a2, a is an EP element if and only if a is one-sided core invertible if and only if a is group invertible.