$J$-prime ideals of commutative rings
Abstract
summary:Let $R$ be a commutative ring with identity, and $J(R)$ denote the Jacobson radical of $R$. This paper introduces $J$-prime ideals, generalizing prime ideals, $n$-ideals, and $J$-ideals. A proper ideal $I$ of $R$ is a $J$-prime ideal if for every $a, b \in R$, $ab \in I$ implies $a\in I+J(R) $ or $b \in I$. We characterize rings in which every proper ideal is $J$-prime, showing that a ring has the property that every proper ideal is $J$-prime if and only if it is a quasi-local ring. Also, we show that (0) is a $J$-prime ideal if and only if the ring is présimplifiable. Furthermore, we examine $J$-prime ideal characteristics in various ring constructions, such as homomorphic image of rings, quotient rings, cartesian product rings, polynomial rings, power series rings, trivial ring extension and amalgamation rings.
Fields of Study & Subjects
Rings
Modules
and Algebras
Fuzzy and Soft Set Theory
Advanced Algebra and Logic
Publication details
- Journal / source
- Czechoslovak Mathematical Journal
- Publisher
- Institute of Mathematics, Czech Academy of Sciences
- Volume
- 76
- Issue
- 2
- Pages
- 525–539
- Publication date
- 27 February 2026
- Language
- EN