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

TL;DR

This paper generalizes classical ideal concepts in commutative rings, introducing $ ext{H}_Y$-ideals and strong $ ext{H}_Y$-ideals, and explores their properties and relations to existing ideals within the spectrum topology.

Contribution

It extends the theory of z-ideals and related ideals by defining and analyzing $ ext{H}_Y$-ideals and strong $ ext{H}_Y$-ideals, unifying and broadening classical concepts.

Findings

Generalization of z-ideals to $ ext{H}_Y$-ideals

Extension of Zariski topology results to new ideal classes

Recognition of similarities and differences among ideal concepts

Abstract

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.