PARTIAL ORDERS ON *-REGULAR RINGS

In this work we consider some new partial orders on *-regular rings. Let A be a *-regular ring, P(A) be the lattice of all projectors in A and mu be a sharp normal normalized measure on P(A). Suppose that (A, rho) is a complete metric *-ring with respect to the rank metric rho on A defined as rho(x, y) = mu(l(x-y)) = mu(r(x-y)), x, y is an element of A, where l(a), r(a) is respectively the left and right support of an element a. On A we define the following three partial orders: a (sic)(s) b double left right arrow b = a + c, a perpendicular to c; a (sic)(l) b double left right arrow l(a)b = a; a (sic)(r) b double left right arrow br(a) = a, a perpendicular to c means algebraic orthogonality, that is, ac = ca = a*c = ac* = 0. We prove that the order topologies associated with these partial orders are stronger than the topology generated by the metric rho. We consider the restrictions of these partial orders on the subsets of projectors, unitary operators and partial isometries of *-regular algebra A . In particular, we show that these three orders coincide with the usual order <= on the lattice of the projectors of *-regular algebra. We also show that the ring isomorphisms of *-regular rings preserve partial orders (sic)(l) and (sic)(r).