Relative z-ideals in

For every two ideals I ⊆ J in C ( X ) , we call I a z J -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 z J -ideal. We have shown that for any ideal J in C ( X ) , the sum of every two z J -ideals is a z J -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 z J -ideal and also for given two ideals I ⊆ J in C ( X ) , a greatest z J -ideal contained in I and the smallest z J -ideal containing I are investigated.