Generalized inverses of a factorization in a ring with involution

Let R   be a ring with an involution ⁎ and p,a,q∈Rp,a,q∈R. In this paper, we investigate the necessary and sufficient conditions for paq   to have a {1,3}{1,3}-inverse (respectively, {1,4}{1,4}-inverse). In particular, the equivalences for the existence of the Moore–Penrose inverse of paq   are obtained. As applications, the existence and expression of the Moore–Penrose inverse of a (2,2,0)(2,2,0) matrix over R are given.