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.

摘要

本文介绍和研究了介于z-理想和z °-理想（= d-理想）之间的一类理想，即z _r-理想。A z _r -ideal 是一个z -ideal，它同时是一个r -ideal（环R中的理想I称为r -ideal，如果对于每个非零除数r ∈ R和每个a ∈ R，ra ∈ I蕴含a ∈ I )。与z的总和相反C ( X ) 中的 -ideals 是z -ideal，z r -ideal 的总和不必_是z r _-ideal。我们证明了C ( X )的每两个z _r理想之和是一个z _r理想当且仅当X是一个准F空间。在C ( X ) 中，每一个z° -ideal 都是一个z _r -ideal，我们刻画了空间X，其反之亦然。我们观察到X是余零补空间当且仅当 C ( X ) 中的每个 (素数) r理想是z理想并且每当 C ( X ) 的每个(素) z理想都是r理想时它等价于X几乎是P空间。使用这些事实证明，当且仅当X是P空间时， C ( X ) 的所有r理想的集合和所有z理想的集合一致。