On m-Closure of Ideals in LBI-Subalgebras

IF 0.7 4区数学 Q2 MATHEMATICS

Bulletin of The Iranian Mathematical Society Pub Date : 2024-07-11 DOI:10.1007/s41980-024-00884-x

Masoumeh Etebar, Mehdi Parsinia, Alireza Salehi

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引用次数: 0

Abstract

Let C(X) be the ring of all continuous real-valued functions on a completely regular Hausdorff space X. A subalgebra A(X) of C(X) is said to be closed under local bounded inversion, briefly an LBI-subalgebra, if for every function f in A(X) that is bounded away from zero on a cozero-set E of X, there exists \(g\in A(X)\) such that \(fg|_E=1\). In this paper, for an LBI-subalgebra A(X) the compactification \(\beta _AX\) of X which is homeomorphic with the structure space of A(X) is investigated. Some properties of \(\beta _AX\) similar to the counterparts in \(\beta X\) and some main differences between these compactifications are given. Using the compactification \(\beta _AX\), we establish an m-closure formula for ideals in a class of LBI-subalgebras which provides a generalization of m-closure of ideals in intermediate algebras of C(X) and \(C_c(X)\). We also investigate a characterization of \(\beta \)-ideals for LBI-subalgebras from which it turns out that m-closed ideals coincide with \(\beta \)-ideals in that class of LBI-subalgebras.

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论 LBI 子代数中的 m 闭合顶点

让 C(X) 是完全正则豪斯多夫空间 X 上所有连续实值函数的环。如果对于 A(X) 中在 X 的零集 E 上离零有界的每个函数 f，都存在 \(g\in A(X)\) 使得 \(fg|_E=1\)，则称 C(X) 的子代数 A(X) 在局部有界反转下是闭合的，简言之，是一个 LBI 子代数。本文研究了对于一个 LBI 子代数 A(X) X 的紧凑化 \(\beta _AX\) 与 A(X) 的结构空间同构。给出了 \(\beta _AX\) 与 \(\beta X\) 类似的一些性质，以及这些压缩之间的一些主要区别。利用紧凑化 \(\beta _AX\)，我们建立了一类 LBI 子代数中理想的 m 闭合公式，这个公式提供了 C(X) 和 \(C_c(X)\ 中间代数中理想的 m 闭合的一般化。）我们还研究了 LBI 子代数的 \(beta \)-理想的特征，由此发现 m-封闭理想与该类 LBI 子代数中的\(beta \)-理想是重合的。

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来源期刊

Bulletin of The Iranian Mathematical Society Mathematics-General Mathematics

CiteScore

1.40

自引率

0.00%

发文量

期刊介绍： The Bulletin of the Iranian Mathematical Society (BIMS) publishes original research papers as well as survey articles on a variety of hot topics from distinguished mathematicians. Research papers presented comprise of innovative contributions while expository survey articles feature important results that appeal to a broad audience. Articles are expected to address active research topics and are required to cite existing (including recent) relevant literature appropriately. Papers are critically reviewed on the basis of quality in its exposition, brevity, potential applications, motivation, value and originality of the results. The BIMS takes a high standard policy against any type plagiarism. The editorial board is devoted to solicit expert referees for a fast and unbiased review process.