CHARACTERIZATIONS AND PROPERTIES OF WEAK CORE INVERSES IN RINGS WITH INVOLUTION

Abstract

In a ring with involution, we first investigate some necessary and sufficient conditions under which Jacobson’s lemma for weak core inverses holds true. Then, we present reverse order laws of weak core inverses and some equivalent conditions under which absorption laws of weak core inverses hold true. Finally, some equivalent characterizations of $a^{\ast }$ commuting with the weak core inverse of $a$ are shown, which improve the relevant result of Zhou et al. (Weak group inverses and partial isometries in proper $\ast$-rings, Linear Multilinear Algebra 70:21 (2022), 6876–6890).