# On $\mathcal{H}_Y$-Ideals

## TL;DR

This paper extends the theory of $\\mathcal{H}_Y$-ideals by introducing fixed and free variants, establishing their properties, and characterizing the compactness of Y and regularity of R through these ideals.

## Contribution

It introduces fixed and free $\\mathcal{H}_Y$-ideals and their relative versions, unifies previous results, and characterizes topological and algebraic properties via these ideals.

## Key findings

- Y is compact iff every proper $\\mathcal{H}_Y$-ideal is fixed.
- Every proper ideal is a relative $\\mathcal{H}_Y$-ideal iff R is regular.
- Characterization of fixed and strong $\\mathcal{H}_Y$-ideals in terms of properties of Y and R.

## Abstract

In this article, we continue the studying of $\mathcal{H}_Y$-ideals. We introducing two notions fixed and free $\mathcal{H}_Y$-ideals as an extension of fixed and free z-ideals in C(X) and relative $\mathcal{H}_Y$-ideals as an extension of relative z-ideals. It has been shown that a large amount of the results of the mentioned papers and generally the papers in the literature about these topics, are special cases of the results of this paper. We prove that Y is compact if and only if every proper $\mathcal{H}_Y$-ideal is a fixed $\mathcal{H}_Y$-ideal; if and only if every proper strong H_Y-ideal is a fixed ideal. Also, we show that every proper ideal is a relative $\mathcal{H}_Y$-ideal, if and only if every proper ideal is a strong $\mathcal{H}_Y$-ideal; if and only if R is regular.