<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation> be a prime ring and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${L}$</EquationSource> <EquationSource Format="MATHML"><math> <mi>L</mi> </math></EquationSource> </InlineEquation> be a non-central Lie ideal of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation>. If <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {F}_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {F}_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">F</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> are two generalized skew derivations of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation> such that <Equation ID="Equ27"> <EquationSource Format="TEX">$\begin{aligned} p \Big (\mathcal {F}_1(XY)-\mathcal {F}_2(X)Y-YX\Big )^n=0 \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi>p</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <msub> <mi mathvariant="script">F</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <msub> <mi mathvariant="script">F</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> <mi>Y</mi> <mo>-</mo> <mi>Y</mi> <mi>X</mi> <msup> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mi>n</mi> </msup> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for all <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$X,Y \in {L}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>∈</mo> <mi>L</mi> </mrow> </math></EquationSource> </InlineEquation>, for some fixed nonzero element <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$p \in \mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mi mathvariant="script">R</mi> </mrow> </math></EquationSource> </InlineEquation> and some fixed <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$n \in \mathbb {Z^+}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <msup> <mi mathvariant="double-struck">Z</mi> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$s_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>s</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>.</p>

Annihilator condition of generalized skew derivations with power values acting on Lie ideals in prime rings

期刊论文
发表日期：

2026年2月24日

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M. Bera,
B. Dhara,
S. Ghosh

摘要

Let $\mathcal {R}$ $R$ be a prime ring and ${L}$ $L$ be a non-central Lie ideal of $\mathcal {R}$ $R$ . If $\mathcal {F}_1$ $F_{1}$ and $\mathcal {F}_2$ $F_{2}$ are two generalized skew derivations of $\mathcal {R}$ $R$ such that $\begin{aligned} p \Big (\mathcal {F}_1(XY)-\mathcal {F}_2(X)Y-YX\Big )^n=0 \end{aligned}$ $\begin{matrix} p (F_{1} (X Y) - F_{2} (X) Y - Y X)^{n} = 0 \end{matrix}$ for all $X,Y \in {L}$ $X, Y \in L$ , for some fixed nonzero element $p \in \mathcal {R}$ $p \in R$ and some fixed $n \in \mathbb {Z^+}$ $n \in Z^{+}$ , then $\mathcal {R}$ $R$ satisfies $s_4$ $s_{4}$ .