  Let $R$ be a commutative ring, $Y\subseteq \mathrm{Spec}(R)$ 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 $b\in I$. A strong $\mathcal{H}_Y$-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 $sz^{\circ}$-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.
