Czechoslovak Mathematical Journal, first online, pp. 1-15
$J$-prime ideals of commutative rings
Mohammed ASSALAMI, Suat KOÇ, Najib MAHDOU, Ünsal TEKİR
Received August 26, 2025. Published online February 27, 2026.
Abstract: 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.
Keywords: $J$-prime ideal; prime ideal; $J$-ideal; $n$-ideal; $r$-ideal; amalgamation; trivial ring extension
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Affiliations: Mohammed Assalami, Laboratory of Modelling and Mathematical Structures, Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202, University S.M. Ben Abdellah Fez, Morocco, e-mail: assalami.mohammed.phd@gmail.com; Suat Koç, Department of Mathematics, Marmara University, Yerleşkesi 34722 Kadiköy, Istanbul, Turkey, e-mail: suat.koc@marmara.edu.tr; Najib Mahdou, Laboratory of Modelling and Mathematical Structures, Department of Mathematics, Faculty of Science and Technology of Fez, Box 2202, University S.M. Ben Abdellah Fez, Morocco, e-mail: mahdou@hotmail.com; Ünsal Tekir (corresponding author), Department of Mathematics, Marmara University, Yerleşkesi 34722 Kadiköy, Istanbul, Turkey, e-mail: utekir@marmara.edu.tr
