Czechoslovak Mathematical Journal, first online, pp. 1-17
On weakly $(1,n)$-ideals and weakly $n$-ideals
Bayram Ali Ersoy, Suat Koç, Ünsal Tekir, Gürsel Yeşilot, Eda Yıldız
Received August 12, 2024. Published online March 28, 2025.
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 $0\neq abc\in I$ for some nonunits $a,b,c\in A$ (or $0\neq ab\in I$ for some $a,b\in A)$, then either $ab\in I$ or $c\in\mathfrak{N}(A)$ (or $a\in I$ or $b\in\mathfrak{N}(A)$, respectively), where $\mathfrak{N}(A)$ 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.
Affiliations: Bayram Ali Ersoy, Department of Mathematics, Yildiz Technical University, 34220, Istanbul, Turkey, e-mail: ersoya@yildiz.edu.tr; Suat Koç, Department of Mathematics, Istanbul Medeniyet University, 34700, Istanbul, Turkey, e-mail: suat.koc@medeniyet.edu.tr; Ünsal Tekir, Department of Mathematics, Marmara University, 34722, Istanbul, Turkey, e-mail: utekir@marmara.edu.tr; Gürsel Yeşilot, Eda Yıldız (corresponding author), Department of Mathematics, Yildiz Technical University, 34220, Istanbul, Turkey, e-mail: gyesilot@yildiz.edu.tr, edyildiz@yildiz.edu.tr
