Let Cc(X) = {f ∈ C(X) : |f (X)| ≤ ℵ0}, C F (X) = {f ∈ C(X) : |f (X)| < ∞}, and Lc(X) = {f ∈ C(X) : C f = X}, where C f is the union of all open subsets U ⊆ X such that |f (U )| ≤ ℵ0, and CF (X) be the socle of C(X) (i.e., the sum of minimal ideals of C(X)). It is shown that if X is a locally compact space, then Lc(X) = C(X) if and only if X is locally scattered. We observe that Lc(X) enjoys most of the important properties which are shared by C(X) and Cc(X). Spaces X such that Lc(X) is regular (von Neumann) are characterized. Similarly to C(X) and Cc(X), it is shown that Lc(X) is a regular ring if and only if it is ℵ0-selfinjective. We also determine spaces X such that Soc Lc(X) = CF (X) (resp.,Ri, where Ri = R for each i, and X has a unique infinite clopen connected subset. The converse of the latter result is also given. The spaces X for which CF (X) is a prime ideal in Lc(X) are characterized and consequently for these spaces, we infer that Lc(X) can not be isomorphic to any C(Y ).

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.

<p>Spaces X for which the annihilator of Sλ(X), the λ-super socle of C(X) (i.e., the set of elements of C(X) that cardinality of their cozerosets are less than λ, where λ is a regular cardinal number such that λ≤|X|) is generated by an idempotent are characterized. This enables us to find a topological property equivalent to essentiality of Sλ(X). It is proved that every prime ideal in C(X) containing Sλ(X) is essential and it is an intersection of free prime ideals. Primeness of Sλ(X) is characterized via a fixed maximal ideal of C(X).</p>