Research Article

On SV_c -Spaces

Published in: Quaestiones Mathematicae
Volume 48 , issue 11 , 2025 , pages: 1619–1639

DOI: 10.2989/16073606.2025.2536069

Author(s): Amir Veisi Yasouj University, Iran , Zahra Keshtkar Yasouj University, Iran

Keywords: Primary: 54C30 , 54C40 , Secondary: 13A18 , Zero-dimensional space , SV_c -space , ring of continuous functions , SV-ring , real closed ring , valuation ring , Bézout ring , DP , pseudoprime ideal , absolutely convex ideal , saturated ideal , strongly irreducible ideal

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Abstract

We say that a topological space X is an SV_c-space if C_c (X) is an SV - ring, that is, is a valuation ring for every prime ideal P of C_c (X). The purpose of this paper is to study and investigate SV_c -spaces. To this end, we will first learn more about real closed and valuation rings. We show that an ordered domain is real closed (resp., satisfies DP) if and only if it is a convex subring of a real closed field (resp., of its field of fractions). After some general results on SV-rings, our attention is focused on the domain . It is shown that for this ring, these four concepts of real closed, DP, valuation, and Bézout are the same, although they do not generally coincide. Several algebraic and topological characterizations of SVc-spaces are given. It is observed that a space X is an SV_c -space if and only if υ ₀ X (υX) is an SV_c -space. Using the notion of -ideals, we obtain: X is an SV_c -space if and only if every prime -ideal (z_c -ideal) of C_c (X) is real closed if and only if every minimal prime ideal of C_c (X) is real closed. The final characterization of SV_c -spaces via ideals of C_c (X) occurs after determining pseudoprime ideals in C_c (X). We first prove that an ideal I of C_c (X) is pseudoprime (quasi primary) if and only if the prime ideals containing I form a chain, and then, it is proven that X is an SV_c -space if and only if every pseudoprime ideal in C_c (X) is strongly irreducible (irreducible) if and only if every pseudoprime ideal in C_c (X) is absolutely convex (convex) if and only if every saturated ideal in C_c (X) is absolutely convex (convex).

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