Boolean algebras

#ring_theory

A Boolean algebra is a ring $A$ with $1$ satisfying $x^{2} = x$ for all $x \in A$ . Prove that in a Boolean algebra $A$ :

$2 x = 0$ for all $x \in A$ .
Every prime ideal $p$ is maximal, and $A / p$ is a field with two elements.

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A {\bfseries Boolean algebra} is a ring $A$ with $1$ satisfying $x^2=x$ for all $x\in A$. Prove that in a Boolean algebra $A$:
	\begin{enumerate}[label=\alph*)]
		\item $2x=0$ for all $x\in A$.
		
		\item Every prime ideal $\mathfrak{p}$ is maximal, and $A/\mathfrak{p}$ is a field with two elements.
	\end{enumerate}