## Finitely Generated Torsion Module Over an Integral Domain Has a Nonzero Annihilator

## Problem 432

**(a)** Let $R$ be an integral domain and let $M$ be a finitely generated torsion $R$-module.

Prove that the module $M$ has a nonzero annihilator.

In other words, show that there is a nonzero element $r\in R$ such that $rm=0$ for all $m\in M$.

Here $r$ does not depend on $m$.

**(b)** Find an example of an integral domain $R$ and a torsion $R$-module $M$ whose annihilator is the zero ideal.