In abstract algebra, a branch of mathematics, a torsion module is a module which, in effect, ignores the action of its ring. In other words, a right R-module M is torsion if, for every element r of R and every element m of M, we have mr=0. R is also said to act trivially on M. The opposite of a torsion module is a torsion-free or faithful module.

There are torsion modules over every ring. Indeed, we can take any abelian group A and give R the trivial action on A—that is, we define ar to be zero for all a in A and all r in R. Then A with this action of R is a torsion module by definition. In fact, all torsion modules arise in this way, because every module has an underlying abelian group, and the condition of being torsion forces R to act trivially.

If R has an identity element, then there is a non-torsion module over R, namely R acting on itself by right multiplication.

Torsion submodules

Torsion is often inconvenient, because it makes the action of the ring useless. If R is commutative, we define the torsion submodule T of M to be the set of all elements of M which are annihilated by some element of R. In other words, T={m | for some r∈R, mr=0}. The quotient M/T is the torsion-free part of M. If we take an element m+T in M/T and an element r in R, we see that (m+T)r=0 if and only if mr is in T. mr in T, however, implies that for some r', mrr'=0, and consequently m was in T to start with. Consequently, M/T is torsion-free.

If we wish to make M torsion-free without taking a quotient (and thereby losing some information), we can consider M as a module over a different ring, namely R modulo the annihilator of M. M is torsion-free as a module over this ring.