Properties of Boolean rings

#ring_theory

Let $A$ be a commutative ring with unit. We call $A$ Boolean if $a^{2} = a$ for every $a \in A$ . Prove that in a Boolean ring $A$ each of the following holds:

$2 a = 0$ for every $a \in A$ .
If $I$ is a prime ideal then $A / I$ is a field with two elements (and in particular $I$ is maximal).
If $I = (a, b)$ is the ideal generated by $a$ and $b$ then $I$ can be generated by the single element $a + b + a b$ . Conclude that every finitely generated ideal is principal.

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Let $A$ be a commutative ring with unit. We call $A$ {\bfseries Boolean} if $a^2=a$ for every $a\in A$. Prove that in a Boolean ring $A$ each of the following holds:
\begin{enumerate}[label=(\alph*)]
	\item $2a=0$ for every $a\in A$.
	\item If $I$ is a prime ideal then $A/I$ is a field with two elements (and in particular $I$ is maximal).
	\item If $I=(a,b)$ is the ideal generated by $a$ and $b$ then $I$ can be generated by the single element $a+b+ab$. Conclude that every finitely generated ideal is principal.
\end{enumerate}