The λ-super socle of the ring of continuous functions

The concept of λ-super socle of C(X), denoted by Sλ (X) (that is, the set of elements of C(X) such that the cardinality of their cozerosets are less than λ, where λ is a regular cardinal number with λ ≤ |X|) is introduced and studied. Using this concept we extend some of the basic results concerning SCF(X), the super socle of C(X) to Sλ(X), where λ ≥ℵ0. In particular, we determine spaces X for which SCF(X) and Sλ(X) coincide. The onepoint λ-compactification of a discrete space is algebraically characterized via the concept of λ-super socle. In fact we show that X is the onepoint λ-compactification of a discrete space Y if and only if Sλ(X) is a regular ideal and Sλ(X)=Ox, for some x ϵ X.