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On \(\mathcal {H}_Y\)-Ideals

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Abstract

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.

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Acknowledgements

I am very grateful to the referees whose valuable suggestions and comments improved an earlier version of this article.

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  1. Jundi-Shapur University of Technology, Dezful, Iran

    Mehdi Badie

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  1. Mehdi Badie
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Correspondence to Mehdi Badie.

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Communicated by Ali Taherifar.

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Badie, M. On \(\mathcal {H}_Y\)-Ideals. Bull. Iran. Math. Soc. 47, 1081–1095 (2021). https://doi.org/10.1007/s41980-020-00429-y

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