This is a question from Dummit/Foote, Chapter 10.3 question 5. I am 100% sure I am misunderstanding this question. Do not write me a solution, I am not asking that.
Let $R$ be a integral domain. Prove every finitely generated torsion module has a nonzero annihilator i.e. there exists nonzero $r\in R $ such that $rm = 0$
The definition of torsion here $M = Tor(M) = \{m : \exists r\neq 0, rm = 0 \} = RA$
Annihilator $Ann(M) = \{ \mu \in R: \mu m = 0,\forall m \in M \} $.
Since $M$ is torsion, every element will have a nonzero $r$ such the $rm = 0$, so $Ann(M) \neq 0$. What am I misunderstanding? What are these additional assumptions for?