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Özen, Mehmet; Naji, Osama A.; Tekir, Ünsal; Koç, Suat

On modules satisfying \(S\)-Noetherian spectrum condition. (English) Zbl 1522.13028

Commun. Math. Stat. 11, No. 3, 649-662 (2023).
In this article, the focus is on modules satisfying the \(S\)-Noetherian spectrum condition. The authors define this condition for a commutative ring with a multiplicatively closed subset and a module. They provide properties and characterizations of modules satisfying this condition and prove a result analogous to Cohen’s theorem. They also characterize modules with a Noetherian spectrum in terms of modules satisfying the \(S\)-Noetherian spectrum condition.
Reviewer: Refat Abdelmawla Khaled Assaad (Rabat)

Cited in 1 Document

MSC:

13E05 Commutative Noetherian rings and modules

Keywords:

Noetherian modules; \(S\)-Noetherian modules; Noetherian spectrum; \(S\)-Noetherian spectrum condition.
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References:

[1] Ahmed, H., S-Noetherian spectrum condition, Commun. Algebra, 46, 8, 3314-3321 (2018) · Zbl 1395.13016 · doi:10.1080/00927872.2017.1412455
[2] Ameri, R., On the prime submodules of multiplication modules, Int. J. Math. Math. Sci., 2003, 27, 1715-1724 (2003) · Zbl 1042.16001 · doi:10.1155/S0161171203202180
[3] Anderson, DD; Winders, M., Idealization of a module, J. Commut. Algebra, 1, 1, 3-56 (2009) · Zbl 1194.13002 · doi:10.1216/JCA-2009-1-1-3
[4] Anderson, DD; Dumitrescu, T., S-Noetherian rings, Commun. Algebra, 30, 9, 4407-4416 (2002) · Zbl 1060.13007 · doi:10.1081/AGB-120013328
[5] Eisenbud, D., Subrings of Artinian and Noetherian rings, Math. Ann., 185, 3, 247-249 (1970) · Zbl 0207.04702 · doi:10.1007/BF01350264
[6] El-Bast, ZA; Smith, PE, Multiplication modules, Commun. Algebra, 16, 4, 755-779 (1988) · Zbl 0642.13002 · doi:10.1080/00927878808823601
[7] Gilmer, R.; Heinzer, W., The Laskerian condition, power series rings and Noetherian spectra, Proc. Am. Math. Soc., 79, 1, 13-16 (1980) · Zbl 0447.13009 · doi:10.1090/S0002-9939-1980-0560575-6
[8] Gilmer, R.: Multiplicative Ideal Theory, Queen’s Papers in Pure and Applied Mathematics, vol. 12. Queen’s University, Kingston (1968) · Zbl 0155.36402
[9] Heinzer, W.; Ohm, J., Locally Noetherian commutative rings, Trans. Am. Math. Soc., 158, 2, 273-284 (1971) · Zbl 0223.13017 · doi:10.1090/S0002-9947-1971-0280472-2
[10] Hedayat, S.; Rostami, E., A characterization of commutative rings whose maximal ideal spectrum is Noetherian, J. Algebra Appl., 17, 1, 1850003 (2018) · Zbl 1387.13014 · doi:10.1142/S0219498818500032
[11] Karakas, HI, On Noetherian modules, METU J. Pure Appl. Sci., 5, 2, 165-168 (1972) · Zbl 0354.13011
[12] Lu, CP, Modules with Noetherian spectrum, Commun. Algebra, 38, 807-828 (2010) · Zbl 1189.13012 · doi:10.1080/00927870802578050
[13] Lu, CP, Prime submodules of modules, Comment. Math. Univ. St. Paul., 33, 61-69 (1984) · Zbl 0575.13005
[14] Lu, CP, Modules and rings satisfying (ACCR), Proc. Am. Math. Soc., 117, 1, 5-10 (1993) · Zbl 0768.13002 · doi:10.1090/S0002-9939-1993-1104398-7
[15] McCasland, RL; Moore, ME, On radicals of submodules of finitely generated modules, Canad. Math. Bull., 29, 1, 37-39 (1986) · Zbl 0546.13001 · doi:10.4153/CMB-1986-006-7
[16] McCasland, RL; Moore, ME, On radicals of submodules, Commun. Algebra, 19, 5, 1327-1341 (1991) · Zbl 0745.13001 · doi:10.1080/00927879108824205
[17] Moore, ME; Smith, SJ, Prime and radical submodules of modules over commutative rings, Commun. Algebra, 30, 10, 5037-5064 (2002) · Zbl 1049.13001 · doi:10.1081/AGB-120014684
[18] Ohm, J.; Pendleton, RL, Rings with Noetherian spectrum, Duke Math. J., 35, 3, 631-639 (1968) · Zbl 0172.32201 · doi:10.1215/S0012-7094-68-03565-5
[19] Park, MH, Noetherian-like properties in polynomial and power series rings, J. Pure Appl. Algebra, 223, 9, 3980-3988 (2019) · Zbl 1419.13035 · doi:10.1016/j.jpaa.2018.12.015
[20] Rush, DE, Noetherian spectrum on rings and modules, Glasgow Math. J., 53, 3, 707-715 (2011) · Zbl 1227.13009 · doi:10.1017/S0017089511000280
[21] Sharp, R.Y.: Steps in Commutative Algebra, vol. 51. Cambridge University Press (2000) · Zbl 0969.13001
[22] Smith, PF, Some remarks on multiplication modules, Archiv der Mathematik, 50, 3, 223-235 (1988) · Zbl 0615.13003 · doi:10.1007/BF01187738
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