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<p>We compute the PI-exponent of the matrix ring with coefficients in an associative algebra. As a consequence, we prove the following. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\mathcal {R}}\)</EquationSource> </InlineEquation> be a PI-algebra with a positive PI-exponent. If <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(M_n({\mathcal {R}})\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(M_m({\mathcal {R}})\)</EquationSource> </InlineEquation> satisfy the same set of polynomial identities then <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n=m\)</EquationSource> </InlineEquation>. We provide examples where this result fails if either <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\({\mathcal {R}}\)</EquationSource> </InlineEquation> is not PI or has zero exponent. We obtain the same statement for certain finite-dimensional algebras with generalized action over an algebraically closed field of zero characteristic.</p>

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On the PI-exponent of matrix algebras and algebras with generalized actions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月3日

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  • Thiago Castilho de Mello,
  • Felipe Yukihide Yasumura

摘要

We compute the PI-exponent of the matrix ring with coefficients in an associative algebra. As a consequence, we prove the following. Let \({\mathcal {R}}\) be a PI-algebra with a positive PI-exponent. If \(M_n({\mathcal {R}})\) and \(M_m({\mathcal {R}})\) satisfy the same set of polynomial identities then \(n=m\) . We provide examples where this result fails if either \({\mathcal {R}}\) is not PI or has zero exponent. We obtain the same statement for certain finite-dimensional algebras with generalized action over an algebraically closed field of zero characteristic.