Talk:Torsion (modules)

The definition of a torsion module that was at torsion module seemed to be quite peculiar. I started a new article as the source I was using defined torsion element-wise, so it seemed silly to keep the title of the article as 'torsion modules'. I don't trust my knowledge enough to write about the non-commative case, or to decide whether the rest of the previous article made sense or not so I have moved it here. Number 0 16:23, 11 January 2006 (UTC)


 * 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&mdash;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&isin;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.