# Tensor product of modules over twisted differential operators

Let $$R$$ be an algebra over complex numbers. Let $$N$$ be a module over $$R$$. We can define the algebra $$D(N)$$ of differential operators $$N \rightarrow N$$ using Grothendieck’s approach as follows: we define it inductively saying that $$D_{-1}(N)=\{0\}$$ and $$D_n(N) \subset \operatorname{End}(N)$$ consist of maps $$d\colon N \rightarrow N$$ such that $$[d,f] \in D_{n-1}(N)$$ for every $$f \in R$$.

Now assume that we are given two $$R$$-modules $$N_1,\, N_2$$ and also let $$M_1,\, M_2$$ be modules over $$D(N_1),\, D(N_2)$$ respectively.

Question: is it true (in this generality) that the tensor product $$M_1 \otimes_R M_2$$ is a module over $$D(N_1 \otimes_R N_2)$$?