This is a follow up of the question Example of a finitely generated faithful torsion module over a commutative ring on MathSE.

Let $M$ be a finitely generated module over a commutative ring $R$ with the property that $\operatorname{Ann}_R x\ne 0$ for all $x \in M$. When $\operatorname{Ann}_R M\ne0$?

The simplest case is $R$ an integral domain. But what about $R$ (local) artinian, or noetherian? (In the counterexample I gave to the linked question $R$ is a commutative ring which is not noetherian.)