On forward-order law for core inverse in rings

TL;DR

This paper investigates conditions under which the core inverse in rings with involution satisfies the forward-order law, extending it to weighted and triple cases, and exploring relations with other generalized inverses.

Contribution

It establishes new sufficient conditions for the forward-order law for the core inverse and introduces related laws involving weighted and hybrid inverses.

Findings

Forward-order law for core inverse established under new conditions

Extension of the law to weighted and triple core inverses

Discussion of hybrid laws among Moore-Penrose, group, and core inverses

Abstract

If $a$ and $b$ are a pair of invertible elements, then $ab$ is also invertible and the inverse of the product $ab$ satisfying $$(ab)^{-1}=a^{-1}b^{-1}$$ is known as the {\it forward-order law}. This article establishes a few sufficient conditions of the forward-order law for the core inverse of elements in rings with involution. It also presents the forward-order law for the weighted core inverse and the triple forward-order law for the core inverse. Additionally, we discuss the hybrid forward-order law among the Moore-Penrose inverse, the group inverse, and the core inverse.