On z_r -ideals of C(X)

Abstract

In this paper we introduce and study a class of ideals between z-ideals and z°-ideals (=d-ideals) namely z_r -ideals. A z_r -ideal is a z-ideal which is at the same time an r-ideal (an ideal I in a ring R is called an r-ideal if for each non-zerodivisor r ∈ R and each a ∈ R, ra ∈ I implies a ∈ I). In contrast to the sum of z-ideals in C(X) which is a z-ideal, the sum of z_r -ideals need not be a z_r -ideal. We prove that the sum of every two z_r -ideals of C(X) is a z_r -ideal if and only if X is a quasi F -space. In C(X) every z°-ideal is a z_r -ideal and we characterize the spaces X for which the converse is also true. We observe that X is a cozero complemented space if and only if every (prime) r-ideal in C(X) is a z-ideal and whenever every (prime) z-ideal of C(X) is an r-ideal it is equivalent to X being an almost P-space. Using these facts it turns out that the set of all r-ideals and the set of all z-ideals of C(X) coincide if and only if X is a P-space.