An extension of z-ideals and z^0-ideals

Abstract

Let R be a commutative ring, Y ⊆ Spec(R) and h Y (S) = {P ∈ Y : S ⊆ P }, for every S ⊆ R. An ideal I is said to be an H Y -ideal whenever it follows from h Y (a) ⊆ h Y (b) and a ∈ I that b ∈ 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, zideal, z • -ideal (d-ideal), sz-ideal and sz • -ideal (ξ-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.