Article
| Title: | A study of $S$-primary decompositions (English) |
| Author: | Singh, Tushar |
| Author: | Ansari, Ajim Uddin |
| Author: | Kumar, Shiv Datt |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 4 |
| Year: | 2025 |
| Pages: | 1241-1253 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | Let $R$ be a commutative ring with identity, and let $S \subseteq R$ be a multiplicative set. An ideal $Q$ of $R$ (disjoint from $S$) is said to be $S$-primary if there exists an $s\in S$ such that for all $x,y\in R$ with $xy\in Q$, we have $sx\in Q$ or $sy\in {\rm rad}(Q)$. Also, we say that an ideal of $R$ is $S$-primary decomposable or has an $S$-primary decomposition if it can be written as a finite intersection of $S$-primary ideals. First we provide an example of an $S$-Noetherian ring in which an ideal does not have a primary decomposition. Then our main aim is to establish the existence and uniqueness of $S$-primary decomposition in $S$-Noetherian rings as an extension of a historical theorem of Lasker-Noether. (English) |
| Keyword: | $S$-Noetherian ring |
| Keyword: | $S$-primary ideal |
| Keyword: | $S$-irreducible ideal |
| Keyword: | $S$-primary decomposition |
| MSC: | 13B02 |
| MSC: | 13C05 |
| MSC: | 13E05 |
| DOI: | 10.21136/CMJ.2025.0029-25 |
| . | |
| Date available: | 2025-12-20T07:20:22Z |
| Last updated: | 2025-12-22 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153240 |
| . | |
| Reference: | [1] Anderson, D. D., Dumitrescu, T.: $S$-Noetherian rings.Commun. Algebra 30 (2002), 4407-4416. Zbl 1060.13007, MR 1936480, 10.1081/AGB-120013328 |
| Reference: | [2] Atiyah, M. F., Macdonald, I. G.: Introduction to Commutative Algebra.Addison-Wesley, Reading (1969). Zbl 0175.03601, MR 0242802 |
| Reference: | [3] Baeck, J., Lee, G., Lim, J. W.: $S$-Noetherian rings and their extensions.Taiwanese J. Math. 20 (2016), 1231-1250. Zbl 1357.16039, MR 3580293, 10.11650/tjm.20.2016.7436 |
| Reference: | [4] Bilgin, Z., Reyes, M. L., Tekir, Ü.: On right $S$-Noetherian rings and $S$-Noetherian modules.Commun. Algebra 46 (2018), 863-869. Zbl 1410.16026, MR 3764903, 10.1080/00927872.2017.1332199 |
| Reference: | [5] Gilmer, R., Heinzer, W.: The Laskerian property, power series rings and Noetherian spectra.Proc. Am. Math. Soc. 79 (1980), 13-16. Zbl 0447.13009, MR 0560575, 10.1090/S0002-9939-1980-0560575-6 |
| Reference: | [6] Hamed, A.: $S$-Noetherian spectrum condition.Commun. Algebra 46 (2018), 3314-3321. Zbl 1395.13016, MR 3788995, 10.1080/00927872.2017.1412455 |
| Reference: | [7] Hamed, A., Hizem, S.: $S$-Noetherian rings of the forms $\Bbb A [X]$ and $\Bbb A [[X]]$.Commun. Algebra 43 (2015), 3848-3856. Zbl 1329.13014, MR 3360852, 10.1080/00927872.2014.924127 |
| Reference: | [8] Hamed, A., Hizem, S.: Modules satisfying the $S$-Noetherian property and $S$-ACCR.Commun. Algebra 44 (2016), 1941-1951. Zbl 1347.13005, MR 3490657, 10.1080/00927872.2015.1027377 |
| Reference: | [9] Hamed, A., Malek, A.: $S$-prime ideals of a commutative ring.Beitr. Algebra Geom. 61 (2020), 533-542. Zbl 1442.13010, MR 4127389, 10.1007/s13366-019-00476-5 |
| Reference: | [10] Kaplansky, I.: Commutative Rings.University of Chicago Press, Chicago (1974). Zbl 0296.13001, MR 0345945 |
| Reference: | [11] Lasker, E.: Zur Theorie der Moduln und Ideale.Math. Ann. 60 (1905), 20-116 German \99999JFM99999 36.0292.01. MR 1511288, 10.1007/BF01447495 |
| Reference: | [12] Lim, J. W.: A note on $S$-Noetherian domains.Kyungpook Math. J. 55 (2015), 507-514. Zbl 1329.13006, MR 3414628, 10.5666/KMJ.2015.55.3.507 |
| Reference: | [13] Massaoud, E.: $S$-primary ideals of a commutative ring.Commun. Algebra 50 (2022), 988-997. Zbl 1481.13007, MR 4379651, 10.1080/00927872.2021.1977939 |
| Reference: | [14] Noether, E.: Idealtheorie in Ringbereichen.Math. Ann. 83 (1921), 24-66 German \99999JFM99999 48.0121.03. MR 1511996, 10.1007/BF01464225 |
| Reference: | [15] Singh, T., Ansari, A. U., Kumar, S. D.: $S$-Noetherian rings, modules and their generalizations.Surv. Math. Appl. 18 (2023), 163-182. Zbl 1537.13035, MR 4657507 |
| Reference: | [16] Vella, D. C.: Primary decomposition in Boolean rings.Pi Mu Epsilon J. 14 (2018), 581-588. Zbl 1432.13004, MR 3887503 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
