Let R be a prime ring with char <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_597_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>, I be a nonzero ideal of R, <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_597_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="96" /> </InlineMediaObject> <EquationSource Format="TEX">$f(x_1,\ldots ,x_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a noncentral multilinear polynomial over extended centroid C and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_597_Article_IEq3.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="79" /> </InlineMediaObject> <EquationSource Format="TEX">$0\ne b'\in R$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≠</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>∈</mo> <mi>R</mi> </mrow> </math></EquationSource> </InlineEquation>. Denote <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_597_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="261" /> </InlineMediaObject> <EquationSource Format="TEX">$f(I)=\{f(t_1,\ldots ,t_n) | t_1,\ldots ,t_n\in I\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo stretchy="false">{</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>t</mi> <mi>n</mi> </msub> <mo>∈</mo> <mi>I</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>. Suppose that F, G and H are three generalized derivations on R such that <Equation ID="Equ30"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_597_Article_Equ30.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="220" /> </MediaObject> <EquationSource Format="TEX">$\begin{aligned} b'\Big \{F\Big (G(V)V\Big )-H(V^2)\Big \}=0 \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <msup> <mi>b</mi> <mo>′</mo> </msup> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">{</mo> </mrow> <mi>F</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <mi>G</mi> <mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> <mi>V</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mo>-</mo> <mi>H</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mi>V</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">}</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for all <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11565_2025_597_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="67" /> </InlineMediaObject> <EquationSource Format="TEX">$V\in f(I)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>V</mi> <mo>∈</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>I</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Then the structure of the maps F, G, H are described. This result naturally generalizes the results obtained by Carini and De Filippis in [Siberian Math. J. 53 (6) (2012), 1051-1060], Dhara and Argac in [Commun. Math. Stat. 4 (2016), 39-54] and completes the incomplete result of Tiwari in [Rend. Circ. Mat. Palermo, II. Ser 71 (2022), 207-223]. At the end of this paper an example is given to show that the primeness condition is not superfluous.

Annihilator conditions and generalized derivations in prime rings

期刊论文
发表日期：

2025年5月25日

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B. Dhara,
S. Kar,
S. Ghosh

摘要

Let R be a prime ring with char $(R)\ne 2$ $(R) \neq 2$ , I be a nonzero ideal of R, $f(x_1,\ldots ,x_n)$ $f (x_{1}, \dots, x_{n})$ be a noncentral multilinear polynomial over extended centroid C and $0\ne b'\in R$ $0 \neq b^{'} \in R$ . Denote $f(I)=\{f(t_1,\ldots ,t_n) | t_1,\ldots ,t_n\in I\}$ $f (I) = {f (t_{1}, \dots, t_{n}) | t_{1}, \dots, t_{n} \in I}$ . Suppose that F, G and H are three generalized derivations on R such that $\begin{aligned} b'\Big \{F\Big (G(V)V\Big )-H(V^2)\Big \}=0 \end{aligned}$ $\begin{matrix} b^{'} {F (G (V) V) - H (V^{2})} = 0 \end{matrix}$ for all $V\in f(I)$ $V \in f (I)$ . Then the structure of the maps F, G, H are described. This result naturally generalizes the results obtained by Carini and De Filippis in [Siberian Math. J. 53 (6) (2012), 1051-1060], Dhara and Argac in [Commun. Math. Stat. 4 (2016), 39-54] and completes the incomplete result of Tiwari in [Rend. Circ. Mat. Palermo, II. Ser 71 (2022), 207-223]. At the end of this paper an example is given to show that the primeness condition is not superfluous.