Suppose R is a non-commutative prime ring with char<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_596_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$(R)\ne 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>≠</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Suppose that <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_596_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="97" /> </InlineMediaObject> <EquationSource Format="TEX">$\kappa \left( x_{1}, \ldots , x_{n}\right) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>κ</mi> <mfenced close=")" open="("> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> </mfenced> </mrow> </math></EquationSource> </InlineEquation> is a noncentral multilinear polynomial over C, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_596_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="272" /> </InlineMediaObject> <EquationSource Format="TEX">$S=\{\kappa (\wp _1,\ldots ,\wp _n) \mid \wp _1,\ldots ,\wp _n \in R\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mo>=</mo> <mo stretchy="false">{</mo> <mi>κ</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>℘</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>℘</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∣</mo> <msub> <mi>℘</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>℘</mi> <mi>n</mi> </msub> <mo>∈</mo> <mi>R</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. If F, G and H are three generalized skew-derivations on R associated to the same automorphism <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_596_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> such that <Equation ID="Equ21"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_596_Article_Equ21.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="170" /> </MediaObject> <EquationSource Format="TEX">$ H\left( \xi \right) \xi -F(\xi )G(\xi )=0 $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi>H</mi> <mfenced close=")" open="("> <mi>ξ</mi> </mfenced> <mi>ξ</mi> <mo>-</mo> <mi>F</mi> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mi>G</mi> <mrow> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </Equation>for each <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_596_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$\xi \in S$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ξ</mi> <mo>∈</mo> <mi>S</mi> </mrow> </math></EquationSource> </InlineEquation>. Let <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_596_Article_IEq6.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$g_1,g_2,g_3 $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>g</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>g</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>g</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> be the associated skew derivations respectively of H, F and G,  such that <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_596_Article_IEq7.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="61" /> </InlineMediaObject> <EquationSource Format="TEX">$g_1,g_2,g_3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>g</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>g</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>g</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> are commuting with <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_596_Article_IEq4.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation>. Then we shall give the structure of H, F and G.

Generalized skew-derivations acting as commuting maps on prime rings

期刊论文
发表日期：

2025年6月20日

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Pallavee Gupta,
S. K. Tiwari

摘要

Suppose R is a non-commutative prime ring with char $(R)\ne 2$ $(R) \neq 2$ . Suppose that $\kappa \left( x_{1}, \ldots , x_{n}\right) $ $κ (x_{1}, \dots, x_{n})$ is a noncentral multilinear polynomial over C, $S=\{\kappa (\wp _1,\ldots ,\wp _n) \mid \wp _1,\ldots ,\wp _n \in R\}$ $S = {κ (℘_{1}, \dots, ℘_{n}) ∣ ℘_{1}, \dots, ℘_{n} \in R}$ . If F, G and H are three generalized skew-derivations on R associated to the same automorphism $\beta $ $β$ such that $ H\left( \xi \right) \xi -F(\xi )G(\xi )=0 $ $H (ξ) ξ - F (ξ) G (ξ) = 0$ for each $\xi \in S$ $ξ \in S$ . Let $g_1,g_2,g_3 $ $g_{1}, g_{2}, g_{3}$ be the associated skew derivations respectively of H, F and G, such that $g_1,g_2,g_3$ $g_{1}, g_{2}, g_{3}$ are commuting with $\beta $ $β$ . Then we shall give the structure of H, F and G.