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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 ?

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    $\begingroup$ See mathoverflow.net/questions/56255/duals-and-tensor-products for related discussion. $\endgroup$
    – Neil Strickland
    Commented Jan 28, 2019 at 10:40
  • $\begingroup$ @NeilStrickland I have done, of course, but my need is to explore what happens in non f.g. modules (I am considering spaces like $k\langle\langle X\rangle\rangle$ (with $X$ a non-void alphabet and $k$ a ring). But, thank you (+1). $\endgroup$
    – Duchamp Gérard H. E.
    Commented Jan 28, 2019 at 11:06

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