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<p>Let <i>R</i> be a ring, <i>P</i> be a prime ideal of <i>R</i>, <i>I</i> be a nonzero ideal of <i>R</i> such that <i>P</i> ⊊ <i>I</i>, <i>D</i><sub>1</sub><i>, D</i><sub>2</sub><i>, D</i><sub>3</sub>: <i>R</i> × <i>R</i> → <i>R</i> be symmetric bi-derivations, and <i>d</i><sub>1</sub><i>, d</i><sub>2</sub><i>,</i> and <i>d</i><sub>3</sub> be the traces of <i>D</i><sub>1</sub><i>, D</i><sub>2</sub> and <i>D</i><sub>3</sub> respectively. We study the following conditions: (i) [<i>d</i><sub>1</sub> (<i>x</i>) <i>x</i>] ∈ <i>P</i>, (ii) <i>d</i><sub>1</sub> (<i>x</i>) ○<i>x</i> ∈ <i>P</i>, (iii) <i>D</i><sub>1</sub> (<i>d</i><sub>2</sub>(<i>x</i>), <i>x</i>) ∈ <i>P</i>, (iv) <i>d</i><sub>1</sub>(<i>d</i><sub>2</sub>)(<i>x</i>)) ± <i>d</i><sub>3</sub> (<i>x</i>) ∈ <i>P</i>, and (v) <i>d</i><sub>1</sub>(<i>x</i>)<i>d</i><sub>2</sub> (<i>x</i>) ∈ <i>P</i> for all <i>x</i>, <i>y</i> ∈ <i>I</i>.</p>

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On Symmetric Bi-Derivations Acting Upon Prime Ideals in Any Rings

  • 期刊论文

  • 发表日期：

    2025年12月16日

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  • Emine Koç Sögütcü,
  • Öznur Gölbaşı,
  • Basudeb Dhara

摘要

Let R be a ring, P be a prime ideal of R, I be a nonzero ideal of R such that P ⊊ I, D1, D2, D3: R × R → R be symmetric bi-derivations, and d1, d2, and d3 be the traces of D1, D2 and D3 respectively. We study the following conditions: (i) [d1 (x) x] ∈ P, (ii) d1 (x) ○x ∈ P, (iii) D1 (d2(x), x) ∈ P, (iv) d1(d2)(x)) ± d3 (x) ∈ P, and (v) d1(x)d2 (x) ∈ P for all x, y ∈ I.