... and prime weak ideals 55 Example 2.4. Let f, g ∈ C(X) such that Z(f) = Z(g) = ∅, f ∈ gC(X), and g ∈ fC(X). We have I = fC(X)∪ gC(X) is a C-weak ideal of C(X), but it is not an ideal of C(X) (see [=-=[4]-=-, Example 1]). It is clear that for a z-weak ideal I of C(X), I is an ideal of C(X) if and only if I is a C-weak ideal of C(X). For every f ∈ C(X), we put Mf = {g ∈ C(X) : Z(f) ⊆ Z(g)} and this notati...