Matrix rings over ∗-regular rings and pseudo-rank functions

Abstract

In this paper we obtain a characterizati on 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. All rings in this paper are associative with 1. Let R be a ring with an involution *. Recall that * is said to be n-positiυe definite if Σ^x X/X* = 0 implies xλ = = xn = 0. The involution * is said to be proper if it is 1-positive definite; and if * is ^-definite positive for all n, then we say that * is positive definite. Recall than an element e e R is said to be a projection if e2 = e* = e and R is called a Rickart *-ring if for every x e R there exists a projection e in R generating the right annihilator of x, that is t(x) = eR. Because of the involution, we have £(x) = Rf for some projection /. Notice that t(x) Π x*R = 0, hence the involution * is proper and R is nonsingular. The above projections e, f depend on x only, 1 — e (1 — f) is called the right (left) projection of x and, as usual, we shall write 1 - e = RP(JC), 1 -/= LP(JC).