Subring structures in C(X) defined by absolute values
Submitted: 2025-06-05
|Accepted: 2025-10-17
|Published: 2026-02-05
Copyright (c) 2025 Mohammad Ali Siavoshi
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
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Keywords:
Algebraic subrings, convex subring, norm-closed subring, norm-reflecting subring, rings of continuous functions
Supporting agencies:
Research Council of Shahid Chamran University of Ahvaz (GN:SCU.MM1403.813)
Abstract:
In this paper, we introduce and study two new classes of subrings of C(X): norm-closed subrings and norm-reflecting subrings. A subring ℝ ⊆ S ⊆ C ( X ) is said to be norm-closed if for every f ∈ S , the function |f| ∈ S; it is norm-reflecting if |f| ∈ S implies f ∈ S. These concepts are inspired by the lattice structure of C(X), particularly the operation of taking absolute values. We provide characterizations of norm-closed (norm-reflecting) subrings. Several examples and counterexamples are presented to illustrate the distinctions and connections between these classes.
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