This will be a very short post about torsion modules. I especially wanted to flesh out an easy characterization of them. I will add more examples eventually…
Let be any commutative ring.
Definition. An -module is said to be torsion-free if implies either that is a zero divisor in , or that .
[TODO Add some examples and non-examples.]
From this point, the ring will always be an integral domain. In this case, the module is torsion-free if and only if the equation implies either that or . Thus we see torsion-freeness as an analog of integrality for rings.
Definition. Given that is an integral domain, the torsion submodule of any -module is the set of elements of that are annihilated by some non-zero element of .
It is really a submodule: if is annihilated by and is annihilated by , then is annihilated by — which is not zero because is an integral domain. For similar reasons, is stable under the action of .
Definition. Given that is an integral domain, an -module is said to be torsion if , that is, if every element of is annihilated by some non-zero element of .
A cheap example of a torsion module is the abelian group for any integer .
Characterization. The module is torsion if and only if
Proof. Suppose first that is torsion, and let be an elementary tensor. Let be a non-zero element such that . Thus On the other hand, suppose is the zero module. We identify this tensor product with , the localization of at the prime ideal . Now any fraction is equal to zero by hypothesis, so there exists some nonzero such that in . Thus , which is an arbitrary element of , lies in . This shows , i.e. is torsion.
In fact, the same reasoning shows that is the kernel of the canonical map This gives an answer to a question I asked myself a while ago. When working with tensor products, if , can we guarantee that ? Well, in our case, that question translates to: is the canonical map an injection? That’s the case if and only if is torsion-free.