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<p>Let <i>R</i> be a noncommutative ring, and let <i>S</i> be an <i>m</i>-system of <i>R</i>. In this paper, we give more results on the concept of almost prime (right) ideals, that were introduced in Abouhalaka and Fındık (Serdica Math J 48:235–246, 2022), particularly, in (right) <i>S</i>-unital rings, local rings, and decomposable rings. In addition, we introduce the concept of almost right <i>S</i>-prime ideals, and we show how some findings regarding almost prime ideals can be derived as corollaries of almost right <i>S</i>-prime ideals. Additionally, we show how almost right <i>S</i>-prime ideals behave in related rings such as homomorphic images, quotient rings, and decomposable rings. Finally, we construct almost right <i>S</i>-prime ideals using the Nagata method of Idealization.</p>

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Generalizations of almost prime and right S-prime ideals in noncommutative rings

  • 期刊论文

  • 发表日期：

    2025年10月17日

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  • Alaa Abouhalaka,
  • Nico Groenewald,
  • Şehmus Fındık

摘要

Let R be a noncommutative ring, and let S be an m-system of R. In this paper, we give more results on the concept of almost prime (right) ideals, that were introduced in Abouhalaka and Fındık (Serdica Math J 48:235–246, 2022), particularly, in (right) S-unital rings, local rings, and decomposable rings. In addition, we introduce the concept of almost right S-prime ideals, and we show how some findings regarding almost prime ideals can be derived as corollaries of almost right S-prime ideals. Additionally, we show how almost right S-prime ideals behave in related rings such as homomorphic images, quotient rings, and decomposable rings. Finally, we construct almost right S-prime ideals using the Nagata method of Idealization.