An extension of $z$-ideals and $z^\circ$-ideals

Year 2020, Volume: 49 Issue: 1, 254 - 272, 06.02.2020

Ali Rezaei Aliabad Mehdi Badie , Sajad Nazari

https://doi.org/10.15672/hujms.455030

Cited By: 6

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

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