Prime z-Ideal Rings (pz-Rings)

A ring R is said to be a pz-ring (resp., psz-ring) whenever every prime ideal of R is a z-ideal (resp., sz-ideal). In this article, we introduce and investigate these concepts. We show that X is a P-space if and only if C(X) is a pz-ring; if and only if C (X) I is a pz-ring for every nonzero ideal I of C(X). Also, we show that if X is a compact space and I is an ideal of C(X), then the quotient ring C (X) I is a pz-ring if and only if I is a finite intersection of maximal ideals. We prove that R[[x]] is never a pz-ring. Also, we introduce a new class of ideals in ∏ λ ∈ Λ R λ denoted by I (F , { I λ }) , where F is a filter on Λ and then we show that R λ is a pz-ring for every λ ∈ Λ if and only if every semiprime ideal of the form I (F , { I λ }) is a z-ideal, where F is an ultrafilter. Moreover, in addition to the main statements of the article, it is proved that an ideal I of R is an sz-ideal if and only if (I, x) is an sz-ideal in R[x]. Also, we show that I is a semiprime ideal in R if and only if I[x] is an sz-ideal in R[x]; if and only I[x] is a z-ideal in R[x]. Using this fact, we present a class of simple examples that shows the sum of two z-ideals is not a z-ideal, in general.