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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.

Reviewer: Jan MalĂ˝ (Praha)

### 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 |