An extension of z-ideals and z°-ideals

Let R be a commutative with unity, Y subset of Spec(R), and h(Y)(S) = {P is an element of Y : S subset of P}, for every S subset of R. An ideal I is said to be an H-Y-ideal whenever it follows from h(Y)(a) subset of h(Y)(b) and a is an element of I that b is an element of I. A strong H-Y-ideal is defined in the same way by replacing an arbitrary finite set F instead of the element a. In this paper these two classes of ideals (which are based on the spectrum of the ring R and are a generalization of the well-known concepts semiprime ideal, z-ideal, z degrees-ideal (d-ideal), sz-ideal and sz degrees-ideal (xi-ideal)) are studied. We show that the most important results about these concepts, Zariski topology, annihilator, and etc. can be extended in such a way that the corresponding consequences seems to be trivial and useless. This comprehensive look helps to recognize the resemblances and differences of known concepts better.