Properties of the annihilator

#ring_theory

Let $R$ be a commutative ring. For each nonempty subset $X \subseteq R$ , the annihilator of $X$ is the set $ann (X) = {a \in R ∣ a x = 0 for all x \in X}$ .

Prove that $ann (X)$ is an ideal of $R$ .
Prove that $X \subseteq ann (ann (X))$ .

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Let $R$ be a commutative ring. For each nonempty subset $X\subseteq R$, the {\bfseries annihilator} of $X$ is the set $\operatorname{ann}(X)=\{a\in R\mid ax=0\text{ for all }x\in X\}$.
\begin{enumerate}[label=\alph*)]
	\item Prove that $\operatorname{ann}(X)$ is an ideal of $R$.
	\item Prove that $X\subseteq \operatorname{ann}(\operatorname{ann}(X))$.
\end{enumerate}