  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.
