Multiplicative Closure of Generalized Moore–Penrose Inverses in Rings with Involution

Let R be a ring with identity equipped with an involution. This paper investigates when the set of all generalized Moore–Penrose inverses of an element is closed under multiplication. We prove that, for every element admitting such inverses, multiplicative closure holds if and only if the element is a self-adjoint idempotent. We further show that, in this case, the inverse set is a monoid isomorphic to the multiplicative monoid of a corner ring. The characterization requires neither an EP assumption nor a proper involution. Matrix examples distinguish projections from EP and non-EP elements.