Notes on realcompactifications generated by ideals

Let X be a Tychonoff space, I be an ideal of C(X) and A(X) be an intermediate algebra of C(X). We show that νIX = νIzX = νIzX = ν X⋃θ(I). Moreover, these equalities are generalized to the ideals of A(X) by investigating a topological characterization for the smallest z-ideal in A(X) containing an ideal J of A(X), Jz(A). Finally, some properties of the mapping ν, which assigns to each subset A of C(X) the raelcompact space νAX, are given