标题： 关于矩阵代数和具有广义作用的代数的PI-指数 显示英文标题

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

摘要： 我们计算了系数在结合代数中的矩阵环的PI指数。 作为结果，我们证明了以下结论。 设 $\mathcal{R}$是一个具有正PI指数的PI代数。 如果 $M_n(\mathcal{R})$和$M_m(\mathcal{R})$满足相同的多项式恒等式，则$n=m$。 我们提供了当$\mathcal{R}$不是PI代数或指数为零时该结果失效的例子。 我们在代数闭域且特征为零的某些有限维代数上得到了相同的结论。 显示英文摘要

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.