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<p>In this study, we define a broader class of ideals in non-commutative rings by considering the special radical of the 2-nil ideals introduced in commutative rings Yetkin Celikel E, Bulletin of the Belgian Mathematical Society Simon Stevin, 28(3) (2021). In the mentioned work, it was shown that in Noetherian integral domain rings, every 2-nil ideal is prime. Here, we demonstrate that in non-commutative rings, every 2-<InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal {P}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">P</mi> </math></EquationSource> </InlineEquation> ideal of an Artinian and prime ring is prime. We also investigated the properties of 2-<InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\varrho \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϱ</mi> </math></EquationSource> </InlineEquation>-ideals in ring constructions such as direct product rings, homomorphic images, and idealization extensions. Moreover, in contrast to the work on commutative rings, some results have also been obtained concerning idealization and we explored the connections of this new class of ideals with various ideal-theoretic structures, including prime ideals, primary-like ideals, and 2-absorbing ideals.</p>

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On 2-\(\varrho \) ideals in noncommutative rings

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年1月16日

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  • Hatice Çay,
  • Nico Groenewald

摘要

In this study, we define a broader class of ideals in non-commutative rings by considering the special radical of the 2-nil ideals introduced in commutative rings Yetkin Celikel E, Bulletin of the Belgian Mathematical Society Simon Stevin, 28(3) (2021). In the mentioned work, it was shown that in Noetherian integral domain rings, every 2-nil ideal is prime. Here, we demonstrate that in non-commutative rings, every 2- \(\mathcal {P}\) P ideal of an Artinian and prime ring is prime. We also investigated the properties of 2- \(\varrho \) ϱ -ideals in ring constructions such as direct product rings, homomorphic images, and idealization extensions. Moreover, in contrast to the work on commutative rings, some results have also been obtained concerning idealization and we explored the connections of this new class of ideals with various ideal-theoretic structures, including prime ideals, primary-like ideals, and 2-absorbing ideals.