ORDERS IN RINGS WITH INVOLUTION

Abstract

In this paper we introduce a definition of order in a (notnecessarily unital) ring with involution in terms of the notions of Moore–Penrose inverse and *-cancellable element instead of those of group inverse and cancellable element. The main result states that if R is a Fountain–Gould order in a ring Q with Q semiprime and coinciding with its socle, then every involution * : R →R can be extended to a (unique) involution on Q in such a way that (R, *) is a *-order in (Q, *). And conversely, every *-order in an involution ring (Q, *) with Q semiprime and coinciding with its socle is a Fountain–Gould order inQ.