On weakly (1,n)-ideals and weakly n-ideals

ERSOY B. A., KOÇ S., TEKİR Ü., YEŞİLOT G., YILDIZ YILMAZ E.

Czechoslovak Mathematical Journal, vol.75, no.2, pp.611-628, 2025 (SCI-Expanded, Scopus) identifier identifier

  • Publication Type: Article / Article
  • Volume: 75 Issue: 2
  • Publication Date: 2025
  • Doi Number: 10.21136/cmj.2025.0337-24
  • Journal Name: Czechoslovak Mathematical Journal
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, DIALNET
  • Page Numbers: pp.611-628
  • Keywords: 1-absorbing primary ideal, 13A15, n-ideal, weakly (1, n)-ideal, weakly n-ideal
  • Yıldız Technical University Affiliated: Yes

Abstract

We study weakly (1, n)-ideals and weakly n-ideals in commutative rings. Let A be a commutative ring with a nonzero identity and I be a proper ideal of A. Then I is said to be a weakly (1, n)-ideal (or weakly n-ideal) if whenever 0 ≠ abc ∈ I for some nonunits a,b,c ∈ A (or 0 ≠ ab ∈ I for some a,b ∈ A), then either ab ∈ I or c∈N(A) (or a ∈ I or b∈N(A), respectively), where N(A) is the set of all nilpotent elements of A. Many examples and properties of weakly (1, n)-ideals and weakly n-ideals are given. We characterize all rings in which every proper ideal is a weakly (1, n)-ideal and weakly n-ideal. Furthermore, we investigate both weakly (1, n)-ideals and weakly n-ideals in amalgamated algebras along an ideal.