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**zelda2139** A commutative ring $\displaystyle A$ is Boolean if $\displaystyle x^2 = x$ for all $\displaystyle x \in A$.

In a Boolean ring $\displaystyle A$, show that every ﬁnitely generated ideal in $\displaystyle A$ is principal.

I am pretty sure this proof is by induction because: $\displaystyle (x,y)=(x-xy+y)$ The inclusion $\displaystyle (\supseteq)$ is clear, and the identities $\displaystyle x(x-xy+y)=x $ and $\displaystyle y(x-xy+y)=y$ give $\displaystyle (\subseteq)$. How do I prove this by induction?