Characterizations for the $(b,c)$-core inverse in rings

Abstract

It is known that group inverse, the Moore--Penrose inverse and the inverse along an element have strongly connections with the classical inverse. The $(b,c)$-inverse and the $(b,c)$-core inverse are two new types of outer generalized inverses, extending several generalized inverses. In this paper, we mainly establish the criteria for the $(b,c)$-inverse and the $(b,c)$-core inverse by units in rings.