Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_580_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{R}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>R</mtext> </math></EquationSource> </InlineEquation> be a noncommutative prime ring equipped with an involution ‘<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_580_Article_IEq2.gif" Format="GIF" Height="9" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX"></EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>’, and let <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_580_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {Q}_{ml}(\textrm{R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">Q</mi> <mrow> <mi mathvariant="italic">ml</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mtext>R</mtext> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the maximal left ring of quotients of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_580_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\textrm{R}$</EquationSource> <EquationSource Format="MATHML"><math> <mtext>R</mtext> </math></EquationSource> </InlineEquation>. The objective of this paper is to characterize additive maps <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_580_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="122" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}:\textrm{R}\rightarrow \mathcal {Q}_{ml}(\textrm{R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo>:</mo> <mtext>R</mtext> <mo stretchy="false">→</mo> <msub> <mi mathvariant="script">Q</mi> <mrow> <mi mathvariant="italic">ml</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mtext>R</mtext> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> that satisfy any one of the following conditions. (i) <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_580_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="287" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}(srs)=\mathcal {H}(s)s^r^+s\mathcal {H}(r)s^+sr\mathcal {H}(s)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mi>r</mi> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi mathvariant="script">H</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>s</mi> <mo>∗</mo> </msup> <msup> <mi>r</mi> <mo>∗</mo> </msup> <mo>+</mo> <mi>s</mi> <mi mathvariant="script">H</mi> <mrow> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <msup> <mi>s</mi> <mo>∗</mo> </msup> <mo>+</mo> <mi>s</mi> <mi>r</mi> <mi mathvariant="script">H</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_580_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$s, r\in \textrm{R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>,</mo> <mi>r</mi> <mo>∈</mo> <mtext>R</mtext> </mrow> </math></EquationSource> </InlineEquation>. (ii) <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_580_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="191" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}(s^s)=\mathcal {H}(s^)s+s^*\mathcal {H}(s)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mo>∗</mo> </msup> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi mathvariant="script">H</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mo>∗</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mi>s</mi> <mo>+</mo> <msup> <mi>s</mi> <mo>∗</mo> </msup> <mi mathvariant="script">H</mi> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_580_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$s\in \textrm{R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>∈</mo> <mtext>R</mtext> </mrow> </math></EquationSource> </InlineEquation>.

Structure of some additive maps in prime rings with involution

期刊论文
发表日期：

2025年2月6日

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Mohammad Aslam Siddeeque,
Abbas Hussain Shikeh,
Raof Ahmad Bhat

摘要

Let $\textrm{R}$ $R$ be a noncommutative prime ring equipped with an involution ‘ $*$ $*$ ’, and let $\mathcal {Q}_{ml}(\textrm{R})$ $Q_{ml} (R)$ be the maximal left ring of quotients of $\textrm{R}$ $R$ . The objective of this paper is to characterize additive maps $\mathcal {H}:\textrm{R}\rightarrow \mathcal {Q}_{ml}(\textrm{R})$ $H : R \to Q_{ml} (R)$ that satisfy any one of the following conditions. (i) $\mathcal {H}(srs)=\mathcal {H}(s)s^*r^*+s\mathcal {H}(r)s^*+sr\mathcal {H}(s)$ $H (s r s) = H (s) s^{*} r^{*} + s H (r) s^{*} + s r H (s)$ for all $s, r\in \textrm{R}$ $s, r \in R$ . (ii) $\mathcal {H}(s^*s)=\mathcal {H}(s^*)s+s^*\mathcal {H}(s)$ $H (s^{*} s) = H (s^{*}) s + s^{*} H (s)$ for all $s\in \textrm{R}$ $s \in R$ .