In this article, we continue studying of \( \mathcal {H}_Y \)-ideals. We introduce notions fixed \( \mathcal {H}_Y \)-ideals, free \( \mathcal {H}_Y \)-ideals and relative \( \mathcal {H}_Y \)-ideals as extensions of fixed z-ideal, free z-ideals and relative z-ideals, respectively. It has been shown that large amounts of the results of 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 \( \mathcal {H}_Y \)-ideal is a fixed \( \mathcal {H}_Y \)-ideal. Also, we show that every proper ideal is a relative \( \mathcal {H}_Y \)-ideal, if and only if every proper ideal is a relative strong \( \mathcal {H}_Y \)-ideal, if and only if R is regular.

在本文中，我们继续研究\（\ mathcal {H} _Y \）- ideals。我们引入固定\（\ mathcal {H} _Y \）-理想，免费\（\ mathcal {H} _Y \）-理想和相对\（\ mathcal {H} _Y \）-理想作为固定z的扩展的概念-理想的自由z理想和相对z理想。已经表明，关于这些主题的文献中大量的论文结果是本文结果的特例。我们证明，当且仅当每个正确的\（\ mathcal {H} _Y \）- ideal是固定的\（\ mathcal {H} _Y \）时，Y才是紧的-理想; 当且仅当每个适当的强\（\ mathcal {H} _Y \）- ideal是固定的\（\ mathcal {H} _Y \）- ideal。此外，我们表明，每一个正确的理想是一个相对\（\ mathcal {H} _Y \） -理想，当且仅当每一个正确的理想是一个相对强大的\（\ mathcal {H} _Y \） -理想，如果和仅当R为规则数时。