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 R be a PI-algebra with a positive PI-exponent. If Mₙ(R) and Mₘ(R) satisfy the same set of polynomial identities then $$n=m$$ . We provide examples where this result fails if either 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.