Characterization of finite Boolean rings

Let R be a finite Boolean ring with 1 \neq 0. Prove that R \cong \mathbb{Z}/(2)^k for some k \geq 1.

We proceed by induction on the cardinality of R. If |R| = 2, then R \cong \mathbb{Z}/(2).

Now suppose that for some k \geq 2, every Boolean ring with identity of cardinality at most k is isomorphic to \mathbb{Z}/(2)^n for some n. Let R be a Boolean ring of cardinality k+1. Now R contains an element e not equal to 1 or 0 which is idempotent (since R is Boolean). By a previous theorem, R \cong Re \times R(1-e). Note that if Re = 0, then e = 0, and likewise if R(1-e) = 0, then e = 1. Thus |Re| and |R(1-e)| are at most k, and we have R \cong \mathbb{Z}/(2)^n \times \mathbb{Z}/(2)^m for some n and m. Thus R \cong \mathbb{Z}/(2)^{n+m}, and the result holds by induction.

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