Let \(R\) be a ring and \(M\) an \(R\)-module. The annihilator of \(M\) is \[ \Ann_R(M) \coloneqq \theset{r\in R\st rm=0 \text{ for every } m\in M}. \]
\(\Ann_R(M)\) is a two-sided ideal of \(R\): it is an additive subgroup, and if \(r\in\Ann_R(M)\) and \(s\in R\), then \((sr)m=s(rm)=0\) and \((rs)m=r(sm)=0\) for every \(m\in M\).
What is the annihilator $\mathrm{ann}_R(M)$ of a module?