Characterizations for the Core Invertibility and EP-ness Involving Projections

In this paper we obtain some equivalent conditions for the core invertibility and EP-ness of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text], [Formula: see text] are projections in different settings, such as ∗-rings, ∗-reducing rings and [Formula: see text]-algebras. Moreover, several representations for the core inverses of product, difference and sum of two generalized projections are derived. In particular, a number of examples are given to illustrate our results.