Annihilators

Suppose $R$ is a ring and $M$ is a left $R$-module.

1. For each submodule $N$ on $M$, the annihilator of $N$ in $R$ is defined to be the set of elements $r\in R$ such that $rn=0$ for all $n\in N$. Prove that the annihilator of $N$ in $R$ is a 2-sided ideal of $R$.
2. For each right ideal $I$ of $R$, the annihilator of $I$ in $M$ is defined to be the set of all elements $m\in M$ such that $im=0$ for all $i\in I$. Prove that the annihilator of $I$ in $M$ is a submodule of $M$.
3. Consider the $\mathbf{Z}$-module $M={\mathbf{Z}}_{24}\times {\mathbf{Z}}_{15}\times {\mathbf{Z}}_{50}$ and ideal $I=2\mathbf{Z}$. Determine the annihilator of $M$ in $\mathbf{Z}$ and the annihilator of $I$ in $M$.