**Question:** Is the following statement true?



> 
> Let $R$ be an associative, commutative, unital ring. Let $M$ and $N$ be $R$-modules. Let $n\geq 1$. Then $Tor\_n^R(M,N)$ is torsion.
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> 


By " $Tor\_n^R(M,N)$ is torsion" I mean that every of its elements is a torsion element. Maybe I want to assume that $R$ is an integral domain.


Remark: The above statement is true if $R$ is a principal ideal domain (then $Tor\_n^R$ vanishes for $n\geq 2$) and $M$ and $N$ are finitely generated (then we can apply the [structure theorem](http://en.wikipedia.org/wiki/Structure_theorem_for_finitely_generated_modules_over_a_principal_ideal_domain)).



Is Tor always torsion?
$Tor$ commutes with extension of scalars, hence (if $R$ is an integral domain and $K$ is its field of fractions), we have
$$
Tor\_n^R(M,N) \otimes\_R K = Tor\_n^K(M\otimes\_R K,N\otimes\_R K).
$$
The right-hand-side vanishes for $n\ge 1$, because $K$ is a field. Hence $Tor$ vanishes after tensoring with $K$, which means that $Tor$ is torsion.


