Let be a ring with and let be a left -module. An element is called torsion if there exists such that . We define .
- Prove that if is an integral domain, then is a submodule of .
- Show by an example that if is not an integral domain, then need not be a submodule.
- Prove that if has zero divisors, then every nontrivial module has nonzero torsion elements.
- We use the submodule criterion. Note that in , so that . Now suppose and . There exist nonzero such that . Since is an integral domain, . Now (since is commutative), so that . Thus is a submodule of .
- Consider the ring , and consider as a left module over itself. Note that neither nor are zero in , but that . Thus . However, , and only if . Thus is not a submodule of , as it is not closed under addition.
- Suppose has zero divisors and let be a left -module. Suppose by way of contradiction that . Now let be nonzero such that , and let be nonzero. Now , so that is torsion. Thus , and so , a contradiction. Thus every nontrivial -module contains nonzero torsion elements.