S-n-ideals of commutative rings

Abstract

Let $R$ be a commutative ring with identity and $S$ a multiplicatively closed subset of $R$. This paper aims to introduce the concept of $S-n$-ideals as a generalization of $n$-ideals. An ideal $I$ of $R$ disjoint with $S$ is called an $S-n$- ideal if there exists $s\in S$ such that whenever $ab \in I$ for $a,~b\in R,$ then $sa\in \sqrt{0}$ or $sb\in I$. The relationships among $S-n$-ideals, $n$-ideals, $S$-prime and $S$-primary ideals are clarified. Besides several properties, characterizations and examples of this concept, S-n-ideals under various contexts of constructions including direct products, localizations and homomorphic images are given. For some particular $S$ and $m\in N$, all $S-n$-ideals of the ring $Z_{m}$ are completely determined. Furthermore, $S-n$-ideals of the idealization ring and amalgamated algebra are investigated.

Keywords

• S-n-ideal
• n-ideal
• S-prime ideal
• S-primary ideal

References

1. Almahdi, F. A., Bouba, E. M., Tamekkante, M. On weakly S-prime ideals of commutative rings, Analele Stiint. ale Univ. Ovidius Constanta Ser. Mat., 29(2) (2021), 173-186. https://doi.org/10.2478/auom-2021-0024
2. Anderson, D. F., Badawi, A., On n-absorbing ideals of commutative rings, Commun. Algebra, 39(5) (2011), 1646–1672. https://doi.org/10.1080/00927871003738998
3. Anderson, D. D., Bataineh, M., Generalizations of prime ideals, Commun. Algebra, 36(2) (2008), 686-696. https://doi.org/10.1080/00927870701724177
4. Anderson, D., Smith, E., Weakly prime ideals, Houst. J. Math., 29(4) (2003), 831-840.
5. Badawi, A., On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc., 75(3) (2007), 417-429. https://doi.org/10.1017/S0004972700039344
6. Darani, A. Y., Generalizations of primary ideals in commutative rings, Novi Sad. J. Math., 42 (2012), 27-35.
7. Calugareanu, G., UN-rings. J. Algebra its Appl., 15(10) (2016), 1650182. https://doi.org/10.1142/S0219498816501826
8. D’Anna, M., Fontana, M., An amalgamated duplication of a ring along an ideal: the basic properties, J. Algebra its Appl., 6(3) (2007), 443–459. https://doi.org/10.1142/S0219498807002326

9. D’Anna, M., Fontana, M., The amalgamated duplication of a ring along a multiplicativecanonical ideal, Ark. Mat., 45(2) (2007), 241-252. https://doi.org/10.1007/s11512-006-0038-1
10. D’Anna, M., Finocchiaro, C. A., Fontana, M., Properties of chains of prime ideals in an amalgamated algebra along an ideal, J. Pure Appl. Algebra, 214 (2010), 1633-1641. https://doi.org/10.1016/j.jpaa.2009.12.008
11. Gilmer, R. W., Multiplicative Ideal Theory, M. Dekker, 1972.
12. Hamed, A., Malek, A., S-prime ideals of a commutative ring, Beitr. Algebra Geom., 61(3) (2020), 533-542. https://doi.org/10.1007/s13366-019-00476-5
13. Khashan, H. A., Bani-Ata, A. B., J-ideals of commutative rings, Int. Electron. J. Algebra, 29 (2021), 148-164. https://doi.org/10.24330/ieja.852139
14. Mohamadian, R., r-ideals in commutative rings, Turkish J. Math., 39(5) (2015), 733-749. https://doi.org/10.3906/mat-1503-35
15. Tekir, U., Koc, S., Oral, K. H., n-ideals of commutative rings, Filomat, 31(10) (2017), 2933-2941. https://doi.org/10.2298/FIL1710933T
16. Visweswaran, S., Some results on S-primary ideals of a commutative ring, Beitr. Algebra Geom., 63(8) (2021), 1-20. https://doi.org/10.1007/s13366-021-00580-5
17. Yassine, A., Nikmehr, M. J., Nikandish, R., On 1-absorbing prime ideals of commutative rings, J. Algebra its Appl., 20(10) (2021), 2150175. https://doi.org/10.1142/S0219498821501759.
18. Yetkin Celikel, E., Generalizations of n-ideals of Commutative Rings, J. Sci. Technol., 12(2) (2019), 650-657. https://doi.org/10.18185/erzifbed.471609