Remarks on the rings of functions which have a finite numb er of di scontinuities

Authors

DOI:

https://doi.org/10.4995/agt.2021.14332

Keywords:

C(X)F, Z-ultrafilter, completely separated, C(X)F -embedded, Z-filter, over-rings of C(X), Artinian ring

Abstract

Let X be an arbitrary topological space. F(X) denotes the set of all real-valued functions on X and C(X)F denotes the set of all f ∈ F(X) such that f is discontinuous at most on a finite set. It is proved that if r is a positive real number, then for any f ∈ C(X)F which is not a unit of C(X)F there exists g ∈ C(X)F such that g ≠ 1 and f = gr f. We show that every member of C(X)F is continuous on a dense open subset of X if and only if every non-isolated point of X is nowhere dense. It is shown that C(X)F is an Artinian ring if and only if the space X is finite. We also provide examples to illustrate the results presented herein.


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Author Biographies

Mohammad Reza Ahmadi Zand, Yazd University

Department of Mathematics

Zahra Khosravi, Yazd University

Department of Mathematics

References

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Z. Gharabaghi, M. Ghirati. and A. Taherifar, On the rings of functions which are discontinuous on a finite set, Houston J. Math. 44, no. 2 (2018), 721-739.

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Published

2021-04-01

How to Cite

[1]
M. R. Ahmadi Zand and Z. Khosravi, “Remarks on the rings of functions which have a finite numb er of di scontinuities”, Appl. Gen. Topol., vol. 22, no. 1, pp. 139–147, Apr. 2021.

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Articles