Is it true that taking injective hull commutes with the tensor product?

Question

Let $M$ and $N$ be two modules (can assume them to be finitely generated if need be) over the ring $A=k[x_0,...,x_n]$. Denote by $E(M)$ the injective hull of $M$. We work in the category of positively graded modules where $\deg x_i=d_i>0$.

My question:

Is it true that $E(E(M)\otimes_A E(N)) \cong E(M)\otimes_A E(N)$? What hypothesis would be needed to achieve this result?