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<p>Let <i>R</i> be a commutative ring with identity and <i>m</i>, <i>n</i> be positive integers. We introduce the class of (<i>m</i>, <i>n</i>)-prime ideals which lies properly between the classes of prime and (<i>m</i>, <i>n</i>)-closed ideals. A proper ideal <i>I</i> of <i>R</i> is called (<i>m</i>, <i>n</i>)-prime if for <i>a</i>, <i>b</i> ∈ <i>R</i>, <i>a</i><sup><i>m</i></sup><i>b</i> ∈ <i>I</i> implies either <i>a</i><sup><i>n</i></sup> ∈ <i>I</i> or <i>b</i> ∈ <i>I</i>. Several characterizations of this new class with many examples are given. Analogous to primary decomposition, we define the (<i>m</i>, <i>n</i>)-decomposition of ideals and show that every ideal in an <i>n</i>-Noetherian ring has an (<i>m</i>, <i>n</i>)-decomposition. Furthermore, the (<i>m</i>, <i>n</i>)-prime avoidance theorem is proved.</p>

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(m, n)-prime ideals of commutative rings

  • 期刊论文

  • 发表日期：

    2025年8月18日

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  • Hani A. Khashan,
  • Ece Yetkin Çelikel

摘要

Let R be a commutative ring with identity and m, n be positive integers. We introduce the class of (m, n)-prime ideals which lies properly between the classes of prime and (m, n)-closed ideals. A proper ideal I of R is called (m, n)-prime if for a, b ∈ R, amb ∈ I implies either an ∈ I or b ∈ I. Several characterizations of this new class with many examples are given. Analogous to primary decomposition, we define the (m, n)-decomposition of ideals and show that every ideal in an n-Noetherian ring has an (m, n)-decomposition. Furthermore, the (m, n)-prime avoidance theorem is proved.