S-PRIME RIGHT SUBMODULES AND AN S-VERSION OF PRIME AVOIDANCE

Abouhalaka A.

Journal of Mathematical Sciences (United States), 2025 (Scopus) identifier

  • Yayın Türü: Makale / Tam Makale
  • Basım Tarihi: 2025
  • Doi Numarası: 10.1007/s10958-025-08015-5
  • Dergi Adı: Journal of Mathematical Sciences (United States)
  • Derginin Tarandığı İndeksler: Scopus, MathSciNet, zbMATH
  • Anahtar Kelimeler: Noncommutative rings, Prime ideals, Prime submodules, Right S-Noetherian rings, S-prime ideals, S-prime submodules
  • Çukurova Üniversitesi Adresli: Evet

Özet

Let S be an m-system of a ring R and P a submodule of a right R-module M. This paper presents the notion of S-prime submodule and provides some properties and equivalent definitions. We define S-multiplication right module and prove that in multiplication (S-multiplication) right R-module M, the ideal (P:RM) is a right S-prime ideal of R if and only if P is an S-prime submodule of M. Moreover, we give an S-version of the prime avoidance lemma. Furthermore, we define S-finite and S-Noetherian right modules following the definitions in Abouhalaka, A (Mediterr. J. Math. 21(2), 43 2024). We prove that a multiplication finitely generated right R-module M is S-Noetherian if (N:RM) is an S-prime ideal of R, for all submodules N of M. In addition, we give some examples of right S-Noetherian rings.