Characterizations of weighted core inverses in rings with involution

Abstract

Let [Formula: see text] be a unital ring with involution. The author investigates the characterizations and representations of weighted core inverse of an element in [Formula: see text] by idempotents and units. Let [Formula: see text], [Formula: see text] be an invertible Hermitian element and [Formula: see text]. We prove that [Formula: see text] is [Formula: see text]-core invertible if and only if there exists an element (or an idempotent) [Formula: see text] such that [Formula: see text], [Formula: see text] and [Formula: see text] (or [Formula: see text]) is invertible. As a consequence, for two invertible Hermitian elements [Formula: see text] and [Formula: see text] in [Formula: see text], [Formula: see text] is weighted-[Formula: see text] with respect to [Formula: see text] if and only if there exists an element (or an idempotent) [Formula: see text] such that [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] (or [Formula: see text]) is invertible. These results generalize and improve conclusions in [T. T. Li and J. L. Chen, Characterizations of core and dual core inverses in rings with involution, Linear Multilinear Algebra 66 (2018) 717–730].