Ünsal TEKİR, Suat KOÇ, Esra ŞENGELEN SEVİM, Tarık ARABACI

On S-prime submodules

In this study, we introduce the concepts of S -prime submodules and S -torsion-free modules, which are generalizations of prime submodules and torsion-free modules. Suppose S ⊆ R is a multiplicatively closed subset of a commutative ring R , and let M be a unital R -module. A submodule P of M with (P :R M ) ∩ S = ∅ is called an S -prime submodule if there is an s ∈ S such that am ∈ P implies sa ∈ (P :R M ) or sm ∈ P. Also, an R -module M is called S -torsion-free if ann(M ) ∩ S = ∅ and there exists s ∈ S such that am = 0 implies sa = 0 or sm = 0 for each a ∈ R and m ∈ M. In addition to giving many properties of S -prime submodules, we characterize certain prime submodules in terms of S -prime submodules. Furthermore, using these concepts, we characterize some classical modules such as simple modules, S -Noetherian modules, and torsion-free modules.

PDF

___

[1] Ahmed H, Sana H. Modules satisfying the S-Noetherian property and S-ACCR. Communications in Algebra 2016; 44 (5): 1941-1951.
[2] Ameri R. On the prime submodules of multiplication modules. International Journal of Mathematics and Mathematical Sciences 2003; 27: 1715-1724.
[3] Anderson DD, Dumitrescu T. S-Noetherian rings. Communications in Algebra 2002; 30 (9): 4407-4416.
[4] Anderson DD, Winders M. Idealization of a module. Journal of Commutative Algebra 2009; 1 (1): 3-56.
[5] Atiyah M. Introduction to Commutative Algebra. Boca Raton, FL, USA: CRC Press, 2018.
[6] Behboodi M, Karamzadeh OAS, Koohy H. Modules whose certain submodules are prime. Vietnam Journal of Mathematics 2004; 32 (3): 303-317.
[7] Bilgin Z, Reyes ML, Tekir Ü. On right S-Noetherian rings and S-Noetherian modules. Communications in Algebra 2018; 46 (2): 863-869.
[8] Darani AY, Mostafanasab H. Co-2-absorbing preradicals and submodules. Journal of Algebra and Its Applications 2015; 14 (7): 1550113.
[9] El-Bast ZA, Smith PF. Multiplication modules. Communications in Algebra 1988; 16 (4): 755-779.
[10] Gilmer R. Multiplicative Ideal Theory. Queen’s Papers in Pure and Applied Mathematics, No. 90. Kingston, Canada: Queen’s University, 1992.
[11] Lu CP. Prime submodules of modules. Commentarii Mathematici Universitatis Sancti Pauli 1984; 33 (1): 61–69.
[12] McCasland RL, Moore ME, Smith PF. On the spectrum of a module over a commutative ring. Communications in Algebra 1997; 25 (1): 79-103.
[13] Nagata M. Local Rings. New York, NY, USA: Interscience, 1962.
[14] Payrovi S, Babaei S. On 2-absorbing submodules. Algebra Colloquium 2012; 19 (1): 913-920.
[15] Sharp RY. Steps in Commutative Algebra (Vol. 51). Cambridge, UK: Cambridge University Press, 2000.
[16] Smith PF. Some remarks on multiplication modules. Archiv der Mathematik 1988; 50 (3): 223-235.
[17] Wang F, Kim H. Foundations of Commutative Rings and Their Modules. Singapore: Springer, 2016.
[18] Zamani N. φ-prime submodules. Glasgow Mathematical Journal 2010; 52 (2): 253-259.

ISSN: 1300-0098
Yayın Aralığı: Yılda 6 Sayı
Yayıncı: TÜBİTAK

Arşiv