- Hacettepe Journal of Mathematics and Statistics
- Vol: 49 Issue: 1
- An extension of $z$-ideals and $z^\circ$-ideals
An extension of $z$-ideals and $z^\circ$-ideals
Authors : Ali Rezaei Aliabad, Mehdi Badie, Sajad Nazari
Pages : 254-272
Doi:10.15672/hujms.455030
View : 11 | Download : 4
Publication Date : 2020-02-06
Article Type : Research
Abstract :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.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