@article{article_455030, title={An extension of $z$-ideals and $z^\circ$-ideals}, journal={Hacettepe Journal of Mathematics and Statistics}, volume={49}, pages={254–272}, year={2020}, DOI={10.15672/hujms.455030}, author={Aliabad, Ali Rezaei and Badie, Mehdi and Nazari, Sajad}, keywords={$z$-ideal,$z^\circ$-ideal,strong $z$-ideal,strong $z^\circ$-ideal,prime ideal,semiprime ideal,Zariski topology,Hilbert ideal,rings of continuous functions}, abstract={<div> Let $R$ be a commutative ring, $Y\subseteq Spec(R)$ 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. </div>}, number={1}, publisher={Hacettepe University}