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On ideals of extension of rings of continuous functions. (English) Zbl 0962.54017

The author considers the family \(\mathcal A \) of all almost continuous functions on a closed interval \(\mathbf I \) (or \(\mathbf I =\mathbf R \)) and the family \(\mathcal C \) of all continuous functions on \(\mathbf I \). Let \(f\in \mathcal A \) be such that the set \(D_f\) of all discontinuity points of \(f\) is closed and contained in \(\{f=0\}\). Then \(\mathcal C _{\mathcal A}\) denotes the \(\mathcal A \)-extension ring of \(\mathcal C \) containing \(f\). If \(\mathcal R \) is a ring, \((f)_{\mathcal R}\) denotes its ideal generated by \(f\). The first theorem shows that there exist a function \(f\in \mathcal A \), a continuum of rings \(\mathcal R _{\eta}\in \mathcal C _{\mathcal A}(f)\) and a continuum of rings \(\mathcal K _{\eta}\in \mathcal C _{\mathcal A}(f)\) such that for \(\eta _1\neq \eta _2\) we have \((f)_{\mathcal R _{\eta _1}}=(f)_{\mathcal R _{\eta _2}}\) whereas \((f)_{\mathcal K _{\eta _1}}=(f)_{\mathcal K _{\eta _2}}\). Also, a further connection between ideals of \(\mathcal C \) and ideals of \(\mathcal R \subset \mathcal C _{\mathcal A}(f)\) is investigated. Porosity and other properties of ideals of \(\mathcal R \subset \mathcal C _{\mathcal A}(f)\) are studied.

MSC:

54C40 Algebraic properties of function spaces in general topology
54C08 Weak and generalized continuity
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
13A15 Ideals and multiplicative ideal theory in commutative rings
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