Abstract
Let [Formula: see text] be a unital ∗-ring. As is well known, idempotents and projections can be constructed by the Moore–Penrose inverse and the core inverse of an element in [Formula: see text]. In this paper, we mainly investigate characterizations and properties of these types of idempotents and projections. Also, several results in [Hartwig and Spindelböck, Matrices for which [Formula: see text] and [Formula: see text] commute, Linear Multilinear Algebra 14 (1984) 241–256] are extended to a general ∗-ring without ∗-cancellable conditions. As applications, the characterization of EP elements is given.
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