Abstract: The Moore-Penrose inverses of matrices over a ring R with an involution * and unitary are discussed. Some necessary and sufficie nt conditions for the existence of the Moore-Penrose inverse of matrices over a ring R are given. In particular, it is obtained that a necessary and sufficient condition for the existence of the Moore-Penrose inverse of a matrix A over a ring R is that A=GDH, where D²=D=D^*, (GD)^{*GD+I-D and DH(DH)*+I-D are invertible.}

Key words: ring, matrix, *-symmetric, Moore-Penrose inverse