On the _x0015_-super socle of C(X)

Suppose that S (X) = f f 2 C(X) : jCoz( f )j = jX n Z( f )j g, where is a regular cardinal number with jXj. The one point -compactification of a -discrete space (i.e, every point of X has been a neighborhood with cardinaity is less than ) is introduced and characterized via the concept of the -super socle. In contrast to this fact that CF(X) (i.e, Socle of C(X)) is never a prime ideal, it is shown that S (X) can be a prime ideal (even a maximal ideal).