Relative z-ideals in C(X)

For every two ideals I⊆J in C(X), we call I a zJ-ideal if Z(f)⊆Z(g), f∈I and g∈J imply that g∈I. An ideal I is called a relative z-ideal, briefly a rez-ideal, if there exists an ideal J such that I⊊J and I is a zJ-ideal. We have shown that for any ideal J in C(X), the sum of every two zJ-ideals is a zJ-ideal if and only if X is an F-space. It is also shown that every principal ideal in C(X) is a rez-ideal if and only if X is an almost P-space and the spaces X for which the sum of every two rez-ideals is a rez-ideal are characterized. Finally for a given ideal I in C(X), the existence of greatest ideal J such that I to be a zJ-ideal and also for given two ideals I⊆J in C(X), a greatest zJ-ideal contained in I and the smallest zJ-ideal containing I are investigated.