One-sided w-core inverses in rings with an involution

This paper contributes to define one-sided versions of ‘w-core inverse’ introduced by the writer. Given any ∗-ring R and a,w∈R, a is called right w-core invertible if there exists some x∈R satisfying awxa = a, awx2=x and awx=(awx)∗. Several characterizations for this type of generalized inverses are given, and it is shown that a is right w-core invertible if and only if a is right w(aw)n−1-core invertible if and only if there exists a Hermitian element p such that pa = 0 and p+(aw)n is right invertible for any integer n≥ 1, in which case, the expression of right w-core inverses is given. Finally, it is proved that right w-core inverses are instances of right inverses along an element, right (b,c)-inverses and right annihilator (b,c)-inverses. As an application, the characterization for the Moore–Penrose inverse is given.