On $S$-pm-rings, $S$-maximal Spectrum, and $S$-clean Rings

Uğur Yiğit

doi:10.47000/tjmcs.1808527

On $S$-pm-rings, $S$-maximal Spectrum, and $S$-clean Rings

Abstract

This study introduces and investigates the idea of $S$-pm-rings, a generalization of pm-rings in the context of commutative rings with a multiplicatively closed subset $S$. We prove that a ring $R$ is an $S$-pm-ring if and only if its $S$-maximal spectrum is a retract (specifically, a deformation retract) of its $S$-prime spectrum. Furthermore, we establish the equivalence of the $S$-pm-ring property to the normality of the $S$-prime spectrum and the Hausdorff property of the $S$-maximal spectrum. We also explore the relationship between $S$-pm-rings and $S$-clean rings, demonstrating that every $S$-local ring is $S$-clean, and every $S$-clean ring is an $S$-pm-ring. These results extend classical theorems in commutative algebra and algebraic geometry to the $S$-version context.

Keywords

References

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Burgess, W.D., Raphael, R., On commutative clean rings and pm rings, in: Rings, Modules and Representations, Contemporary Mathematics, Vol. 480, American Mathematical Society, Providence, RI, 2009, 35–55.
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Details

Primary Language

English

Subjects

Algebra and Number Theory, Topology

Journal Section

Research Article

Authors

Uğur Yiğit ^*
0000-0002-6173-5727
Türkiye

Publication Date

February 23, 2026

Submission Date

October 22, 2025

Acceptance Date

December 5, 2025

Published in Issue

Year 2026 Volume: 18 Number: 1

DOI

https://doi.org/10.47000/tjmcs.1808527

IZ

https://izlik.org/JA48PH62WH

Cite

RIS / Bibtex

APA

Yiğit, U. (2026). On $S$-pm-rings, $S$-maximal Spectrum, and $S$-clean Rings. Turkish Journal of Mathematics and Computer Science, 18(1), 209-215. https://doi.org/10.47000/tjmcs.1808527

AMA

1.Yiğit U. On $S$-pm-rings, $S$-maximal Spectrum, and $S$-clean Rings. TJMCS. 2026;18(1):209-215. doi:10.47000/tjmcs.1808527

Chicago

Yiğit, Uğur. 2026. “On $S$-Pm-Rings, $S$-Maximal Spectrum, and $S$-Clean Rings”. Turkish Journal of Mathematics and Computer Science 18 (1): 209-15. https://doi.org/10.47000/tjmcs.1808527.

EndNote

Yiğit U (February 1, 2026) On $S$-pm-rings, $S$-maximal Spectrum, and $S$-clean Rings. Turkish Journal of Mathematics and Computer Science 18 1 209–215.

IEEE

[1]U. Yiğit, “On $S$-pm-rings, $S$-maximal Spectrum, and $S$-clean Rings”, TJMCS, vol. 18, no. 1, pp. 209–215, Feb. 2026, doi: 10.47000/tjmcs.1808527.

ISNAD

Yiğit, Uğur. “On $S$-Pm-Rings, $S$-Maximal Spectrum, and $S$-Clean Rings”. Turkish Journal of Mathematics and Computer Science 18/1 (February 1, 2026): 209-215. https://doi.org/10.47000/tjmcs.1808527.

JAMA

1.Yiğit U. On $S$-pm-rings, $S$-maximal Spectrum, and $S$-clean Rings. TJMCS. 2026;18:209–215.

MLA

Yiğit, Uğur. “On $S$-Pm-Rings, $S$-Maximal Spectrum, and $S$-Clean Rings”. Turkish Journal of Mathematics and Computer Science, vol. 18, no. 1, Feb. 2026, pp. 209-15, doi:10.47000/tjmcs.1808527.

Vancouver

1.Uğur Yiğit. On $S$-pm-rings, $S$-maximal Spectrum, and $S$-clean Rings. TJMCS. 2026 Feb. 1;18(1):209-15. doi:10.47000/tjmcs.1808527