S-Prime Ideals in Principal Domain

Mohamed Aqalmoun

doi:10.22342/jims.29.1.1201.93-98

Mohamed Aqalmoun ⁽¹⁾

(1) Department of Mathematics, Higher Normal School, Sidi Mohamed Ben Abdellah University, Fez, Morocco

https://doi.org/10.22342/jims.29.1.1201.93-98

Issue
VOLUME 29 NUMBER 1 (MARCH 2023)

Submitted
April 18, 2022

Accepted
February 22, 2023

Published
March 30, 2023

Keywords:

Principal domain, S-prime ideal, S-maximal ideal, S-radical

PDF

Abstract

Let R be a commutative ring and S be a multiplicative subset of R. The S-prime ideal is a generalization
of the concept of prime ideal. In this paper, we completely determine all S-prime and S-maximal ideals
of a principal domain. It is shown that the intersection of any descending chain of S-prime ideals in a
principal domain is an S-prime ideal, also the S-radical is investigated.

Full text article

Generated from XML file

References

Anderson, D. D., A note on minimal prime ideals, Proceedings of the AMS. Vol. 122 (1), 1994.

Anderson, D.D., Dumitrescu, T.: S-Noetherian rings. Commun. Algebr. 30, 44074416 (2002).

Atiyah, M., McDonald, I. G. (2018). Introduction to Commutative Algebra. Oxford: Addison-Wesley Publishing Company.

Eda Yldz, Bayram Ali Ersoy, nsal Tekir and Suat Ko (2020): On S-Zariski topology, Communications in Algebra

Hamed, A., and Malek, A. (2019). S-prime ideals of a commutative ring. Beitrge Zur Algebra Und Geometrie.

Hochster, M. (1971). The minimal prime spectrum of a commutative ring. Can. J. Math. 23(5):749758. DOI:10.4153/CJM-1971-083-8.

Lim, J.W.: A note on S-Noetherian domains. Kyungpook Math. J.55, 507514 (2015)

Liu, Z., On S-Noetherian rings. Arch. Math. (Brno) 43, 5560 (2007)

McCoy, N.H., Rings and ideals. Carus Math. Monogr. 8, 96107 (1948)

Ohm, J., Pendleton, R., Rings with Noetherian spectrum. Duke Math. J. 35:631639, (1968).

Sevim, E. S., Arabaci, T., Tekir, U., Ko, S. (2019). ˙ On S-prime submodules. Turk. J. Math. 43(2):10361046.

Thomas W. Hungerford. On the structure of principal ideal rings. Pacific J. of math., Vol. 25, No. 3, 1968

Wiegand, R., Wiegand, S. Prime ideals in Noetherian rings. Surv. Trend. Math. 13, 175193 (2010)

Wiegand, S., Intersections of prime ideals in Noetherian rings. Comm. Algebr. 11, 18531873 (1983)

O. Zariski and P. Samuel, Commutative Algebra, volume I, Van Nostrand, Princeton, 1960

Authors

Mohamed Aqalmoun

Department of Mathematics, Higher Normal School, Sidi Mohamed Ben Abdellah University, Fez

maqalmoun@yahoo.fr (Primary Contact)

Aqalmoun, M. (2023). S-Prime Ideals in Principal Domain. Journal of the Indonesian Mathematical Society, 29(1), 93–98. https://doi.org/10.22342/jims.29.1.1201.93-98