Let R be prime ring with characteristic different from 2, C denotes the extended centroid, L a Lie ideal of R and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_581_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">$Q_r$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mi>r</mi> </msub> </math></EquationSource> </InlineEquation> the right Martindale quotient of the ring R. Let <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_581_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_581_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> represents two generalized skew derivations of R associated with <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_581_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$(\psi ,l_1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo>,</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_581_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">$(\psi , l_2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo>,</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, respectively, such that <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_581_Article_IEq6.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$\psi .l_1=l_1.\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo>.</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>l</mi> <mn>1</mn> </msub> <mo>.</mo> <mi>ψ</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_581_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="82" /> </InlineMediaObject> <EquationSource Format="TEX">$\psi . l_2= l_2.\psi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo>.</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>l</mi> <mn>2</mn> </msub> <mo>.</mo> <mi>ψ</mi> </mrow> </math></EquationSource> </InlineEquation>. If, for every <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_581_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$r \in L$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>∈</mo> <mi>L</mi> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_581_Article_IEq9.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="118" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _1^2(r)r=\Delta _2(r^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="normal">Δ</mi> <mn>1</mn> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mi>r</mi> <mo>=</mo> <msub> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>r</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, then we characterize the maps <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_581_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_581_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>. As an application of this generalization, we proved that if <InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_581_Article_IEq12.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">$\Delta _1(\tau ^2)=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>τ</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_581_Article_IEq13.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$\tau \in R$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>τ</mi> <mo>∈</mo> <mi>R</mi> </mrow> </math></EquationSource> </InlineEquation>, then R contains a non-zero central ideal.

Generalized skew derivations on Lie ideals in prime rings

期刊论文
发表日期：

2025年3月1日

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Giovanni Scudo,
Ashutosh Pandey,
Balchand Prajapati

摘要

Let R be prime ring with characteristic different from 2, C denotes the extended centroid, L a Lie ideal of R and $Q_r$ $Q_{r}$ the right Martindale quotient of the ring R. Let $\Delta _1$ $Δ_{1}$ and $\Delta _2$ $Δ_{2}$ represents two generalized skew derivations of R associated with $(\psi ,l_1)$ $(ψ, l_{1})$ and $(\psi , l_2)$ $(ψ, l_{2})$ , respectively, such that $\psi .l_1=l_1.\psi $ $ψ . l_{1} = l_{1} . ψ$ and $\psi . l_2= l_2.\psi $ $ψ . l_{2} = l_{2} . ψ$ . If, for every $r \in L$ $r \in L$ , $\Delta _1^2(r)r=\Delta _2(r^2)$ $Δ_{1}^{2} (r) r = Δ_{2} (r^{2})$ , then we characterize the maps $\Delta _1$ $Δ_{1}$ and $\Delta _2$ $Δ_{2}$ . As an application of this generalization, we proved that if $\Delta _1(\tau ^2)=0$ $Δ_{1} (τ^{2}) = 0$ for all $\tau \in R$ $τ \in R$ , then R contains a non-zero central ideal.