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Keywords:
$J$-prime ideal; prime ideal; $J$-ideal; $n$-ideal; $r$-ideal; amalgamation; trivial ring extension
$J$-prime ideal; prime ideal; $J$-ideal; $n$-ideal; $r$-ideal; amalgamation; trivial ring extension
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.
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