S-Prime Ideals, S-Noetherian Noncommutative Rings, and the S-Cohen’s Theorem

Abouhalaka A.

Mediterranean Journal of Mathematics, cilt.21, sa.2, 2024 (SCI-Expanded, Scopus) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 21 Sayı: 2
  • Basım Tarihi: 2024
  • Doi Numarası: 10.1007/s00009-023-02578-w
  • Dergi Adı: Mediterranean Journal of Mathematics
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, DIALNET
  • Anahtar Kelimeler: 16N60, 16W99, Noncommutative rings, Prime ideals, S-Cohen’s theorem, S-Prime ideals
  • Çukurova Üniversitesi Adresli: Evet

Özet

Let S be an m-system of a ring R. This paper presents the notion of a right S-prime ideal into noncommutative rings and provides some properties and equivalent definitions. We define a right S-idempotent ideal and an S-totally ordered set, and we show that every ideal of R is a right S-idempotent ideal, and the set of ideals in R is S-totally ordered if and only if every ideal in R is a right S-prime ideal. We also generalize the concept of an S-finite ideal and an S-Noetherian ring. Furthermore, we provide the S-versions of Cohen’s and Cohen–Kaplansky’s Theorems in a special case, and we demonstrate that the ring T2(R) of upper triangular matrices over an S-Noetherian ring R is right ST2(R)-Noetherian.