In this paper we obtain a characterization of those *-regular rings whose matrix rings are *-regular satisfying LP-* RP. This result allows us to obtain a structure theorem for the *-regular self-injective rings of type I which satisfy LP-* RP matricially. Also, we are concerned with pseudo-rank functions and their corresponding metric completions. We show, amongst other things, that the LP-* RP axiom extends from a unit-regular *-regular ring to its completion with respect to a pseudo-rank function. Finally, we show that the property LP-* RP holds for some large classes of *-regular self-injective rings of type II. © 1987 by Pacific Journal of Mathematics.