The classical book on commutative algebra Introduction to Commutative Algebra, by Atiyah and Macdonald, has the following as exercise I.11.
A ring is Boolean if $x^2=x$ for any $x$ of $A$. In a Boolean ring $A$, show that
i) $2x=0$ for all $x\in A$;
ii) Every prime ideal of $A$ is maximal, and its residue field consists of two elements;
iii) Every finitely generated ideal of $A$ is principal.
I can but solve the first two of them; for the last one, I cannot even perceive the situation here, for it will not, generally speaking, be a Dedekind domain, whereof my knowledge is somewhat more. I have hitherto tried to find one element for each ideal of two generators which can stand the stead of them, and then proceed by induction. But even the first step fails to produce any result.
Thanks for any hints and inspirations in advance.