Prove that in a Boolean ring, every finitely generated ideal is principal.
We begin with a lemma.
Lemma: Let be a ring. Suppose that for all with , is principal. Then for all finite nonempty , is principal. Proof: We proceed by induction on . For the base case, If is 1 or 2, then is principal. For the inductive step, suppose that for some , for all with , is principal. Let have cardinality and write , where . Now , and by the induction hypothesis, for some . Now for some by hypothesis.
Now let be a Boolean ring, and let . We claim that . It is clear that since has a 1 and is commutative. Moreover, note that and , so that . Thus every subset of order 2 generates a principal ideal in , and by the lemma, every finitely generated ideal in is principal.