I've been struggling with the following problem for a couple of days and I don't seem to get any further:
Let $R$ be a commutative ring. I would like to get (something like) a classification of all finitely generated $R$-modules $M$ that satisfy the following condition:
When we look at $M \otimes_R M \otimes_R M$, the permutation (123) induces an automorphism of $M \otimes_R M \otimes_R M$ by sending $a \otimes b \otimes c$ to $c \otimes a \otimes b$. I demand that this automorphism be the identity map. In other words, in $M \otimes_R M \otimes_R M$, the elements $a \otimes b \otimes c, c \otimes a \otimes b, b \otimes c \otimes a$ should all be the same.
If $R$ is a field, it is easy to see that the only non-trivial finitely generated $R$-modules (i.e. finite-dimensional vector spaces) that satisfy this condition are the 1-dimensional ones. Furthermore, one sees more generally that all cyclic modules satisfy the condition, too. Up to now I've neither come up with an example of a non-cyclic module that satisfies the condition, nor was I able to prove that all modules that satisfy this condtion must be cyclic.
Can somebody help with this matter?