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

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.