On the locally socle of C(X) whose local cozeroset is cocountable (cofinite)

Abstract

Let CF (X) be the socle of C(X) (i.e., the sum of minimal ideals of C(X)). We introduce and study the concept of colocally socle of C(X) as CµS λ (X) = f ∈ C(X) : |X\S λ f | < µ , where S λ f is the union of all open subsets U in X such that |U \Z(f)| < λ. CµS λ (X) is a z-ideal of C(X) containing CF (X). In particular, C ℵ 0 S ℵ 0 (X) = CCF (X) and C ℵ 1 S ℵ 1 (X) = CSc(X) are investigated. For each of the containments in the chain CF (X) ⊆ CCF (X) ⊆ CµS λ (X) ⊆ C(X), we characterize the spaces X for which the containment is actually an equality. We determine the conditions such that CCF (X) (CSc(X)) is not prime in any subrings of C(X) which contains the idempotents of C(X). The primeness of CCF (X) in some subrings of C(X) is investigated.