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<p>Let <i>R</i> be a commutative ring with identity, and <i>J</i>(<i>R</i>) denote the Jacobson radical of <i>R</i>. This paper introduces <i>J</i>-prime ideals, generalizing prime ideals, <i>n</i>-ideals, and <i>J</i>-ideals. A proper ideal <i>I</i> of <i>R</i> is a <i>J</i>-prime ideal if for every <i>a, b</i> ∈ <i>R, ab</i> ∈ <i>I</i> implies <i>a</i> ∈ <i>I</i> + <i>J</i>(<i>R</i>) or <i>b</i> ∈ <i>I</i>. We characterize rings in which every proper ideal is <i>J</i>-prime, showing that a ring has the property that every proper ideal is <i>J</i>-prime if and only if it is a quasilocal ring. Also, we show that (0) is a <i>J</i>-prime ideal if and only if the ring is présimplifiable. Furthermore, we examine <i>J</i>-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.</p>

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J-prime ideals of commutative rings

  • 期刊论文

  • 发表日期：

    2026年2月27日

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  • Mohammed Assalami,
  • Suat Koç,
  • Najib Mahdou,
  • Ünsal Tekır

摘要

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 ∈ R, ab ∈ I implies a ∈ I + J(R) or b ∈ 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 quasilocal 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.