# Tensor square of duals over a domain

The title is motivated by my needs ($$M=N$$ in the sequel).

Linked to the question here and there (in the case of products) is the following.

Let $$M,N$$ $$k$$-modules ($$k$$ a commutative ring), then we consider the canonical map $$\Phi:M^*\otimes_k N^*\longrightarrow (M\otimes_k N)^*$$ defined by $$\Phi(\phi^{(1)}\otimes \phi^{(2)})[u\otimes v]=\phi^{(1)}(u)\phi^{(2)}(v)$$

I came across the following statement

Proposition Let $$M,N$$ be modules over $$k$$, an integral domain, then if $$M^*\otimes_k N^*$$ is torsion-free, the canonical map $$\Phi$$ is into.

Question 1 Is it known ? (I have a proof - essentially the same as in there - but no reference).

Question 2 The example of Jeremy Rickard in the question above showed us that we cannot get rid of the hypothesis [$$M^*\otimes_k N^*$$ is torsion-free] in the case of domains, under what additional requirements on $$k$$ (UFD,PID ...) can we get rid of it, then ?