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

Let $R$ be a commutative ring, $Y\subseteq \mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}(R)$ and $h_Y(S)=\{P\in Y:S\subseteq P\}$, for every $S\subseteq R$. An ideal $I$ is said to be an ${\mathcal{H}}_Y$-ideal whenever it follows from $h_Y(a)\subseteq h_Y(b)$and$a\in I$that$b\in I$\mathrm{.}Astrong$$\mathcal{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, $z^{\circ }$-ideal (d-ideal), sz-ideal and $s{z}^{\circ }$-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.