Functional identities on tensor products of algebras

Let R and S be unital algebras. We show that if X is a d-free subset of R and S   is finite dimensional, then the set X={x⊗s|x∈X,s∈S} is a d  -free subset of the algebra R⊗SR⊗S. The assumption that S   is finite dimensional turns out to be necessary in general. However, we show that some important functional identities have only standard solutions on XX even when S is infinite dimensional.