On *-DMP inverses in a ring with involution
Huanyin Chen, Honglin Zou · Communications in Algebra · 2021
An element a in a *-ring R has *-DMP inverse if there exists m∈N such that am∈R#∩R† and (am)#=(am)†. If m = 1, then a is said to have *-gMP inverse. In this paper, we present several new characterizations of *-DMP inverses in a *-ring. We prove that a has *-DMP inverse if and only if there exists a projection e∈comm(a) such that a+e∈R−1 and ae is nilpotent if and only if there exist m∈N and x∈R such that xam+1=am and ax=xa=(ax)*. Moreover, we prove that a∈R has *-gMP inverse if and only if aa*=ua,a*a=av for some u,v∈R−1. The corresponding new characterization of *-DMP inverse is thereby obtained.