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**DanielThrice** I'm just starting work on Boolean rings. I understand so far that if we have some commutative Boolean ring $\displaystyle $ R$$ with unity, that for every $\displaystyle \alpha\in R$ we have $\displaystyle \alpha^2 = \alpha$.

I would think then that some ideal $\displaystyle I$ in $\displaystyle R$ would make the factor ring $\displaystyle R/I$ also boolean. How can I prove that this factor ring $\displaystyle R/I$ is also boolean?

As for maximals, how is it we would prove that every proper nontrivial prime ideal $\displaystyle I<R$ is maximal?