The reverse order law of the $(b, c)$-inverse in rings

TL;DR

This paper investigates the reverse order law for the $(b,c)$-inverse in rings, providing equivalent conditions, studying mixed laws, and extending to more general cases, thus advancing the theoretical understanding of generalized inverses.

Contribution

It introduces new equivalent conditions and extends the reverse order law for the $(b,c)$-inverse in rings, including mixed and more general cases.

Findings

Established equivalent conditions for the reverse order law.

Studied various mixed-type reverse order laws.

Extended results to the inverse along an element and general cases.

Abstract

We present equivalent conditions of reverse order law for the $(b, c)$-inverse $(aw)^{(b,c)}=w^{(b,s)}a^{(t,c)}$ to hold in a ring. Also, we study various mixed-type reverse order laws for the $(b,c)$-inverse. As a consequence, we get results related to the reverse order law for the inverse along an element. More general case of reverse order law $(a_1a_2)^{(b_3, c_3)}=a_2^{(b_2, c_2)}a_1^{(b_1, c_1)}$ is considered too.