On locally socle and its extension

We introduce and study the concept of the locally super socle of C(X), denoted by LS F(X) as LS F(X) = n f 2 C(X) : S f = X o , where S f is the union of all open subsets U in X such that UnZ( f ) is as most countable. LS F(X) is a z-ideal of C(X) containing SCF(X) (i.e., the set of elements of C(X), which are zero everywhere except on a countable number of points of X). We characterize spaces X for which the equality in the relation SCF(X) LS F(X) C(X) is hold. In fact, we show that X is an almost countable discrete space if and only if LS F(X) = C(X). We note that if X is an infinite space, then CF(X) ( C(X). We also observe that jI(X)j @0 if and only if SCF(X) = LS F(X). We determine the conditions such that LS F(X) is not prime in any subrings of C(X) which contains the idempotents of X. We investigate the primness of LCF(X) in some subrings of C(X).