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

Abstract

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 generalizes the recent result about the inverse along an element by L. Wang and D. Mosic [Linear Multilinear Algebra, https://doi.org/10.1080/03081087.2019.1679073 ], 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 studied by J. Chen, H. Zou, H. Zhu, and P. Patricio [Mediterr J. Math., 2017, 14: 208] 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.