In the present article, we propose the idea of Lie Armendariz rings and Jordan Armendariz rings as a generalization of reduced rings and examine their properties. Our findings indicate that both the classes of Lie and Jordan Armendariz rings form a proper subclass of central Armendariz rings while containing the class of reduced rings. We prove that the Laurent polynomial ring <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$R[t,t^{-1}]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo stretchy="false">[</mo> <mi>t</mi> <mo>,</mo> <msup> <mi>t</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> and the polynomial ring R[t] over a Lie Armendariz ring are Lie Armendariz. Furthermore, we show that if R is reduced, then its trivial extension T(R, R) and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$R[t]/(t^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is Lie Armendariz, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(t^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is the ideal generated by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$t^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>t</mi> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Additionally, we prove that direct product of Lie Armendariz rings is Lie Armendariz. Finally, we show that R is reversible if R is Lie Armendariz or Jordan Armendariz, and the converse holds if R is Armendariz, thereby proving that symmetric rings are not the only class of rings between reduced rings and reversible rings. We obtain the similar results for Jordan Armendariz rings.

On Lie and Jordan Armendariz rings

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发表日期：

2025年9月1日

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Asma Ali,
Shervin Sahebi,
Shafahat Hussain

摘要

In the present article, we propose the idea of Lie Armendariz rings and Jordan Armendariz rings as a generalization of reduced rings and examine their properties. Our findings indicate that both the classes of Lie and Jordan Armendariz rings form a proper subclass of central Armendariz rings while containing the class of reduced rings. We prove that the Laurent polynomial ring $R[t,t^{-1}]$ $R [t, t^{- 1}]$ and the polynomial ring R[t] over a Lie Armendariz ring are Lie Armendariz. Furthermore, we show that if R is reduced, then its trivial extension T(R, R) and $R[t]/(t^n)$ $R [t] / (t^{n})$ is Lie Armendariz, where $(t^n)$ $(t^{n})$ is the ideal generated by $t^n$ $t^{n}$ and $n\ge 2$ $n \geq 2$ . Additionally, we prove that direct product of Lie Armendariz rings is Lie Armendariz. Finally, we show that R is reversible if R is Lie Armendariz or Jordan Armendariz, and the converse holds if R is Armendariz, thereby proving that symmetric rings are not the only class of rings between reduced rings and reversible rings. We obtain the similar results for Jordan Armendariz rings.