Kleene S. C.. A note on recursive functions. Bulletin of the American Mathematical Society, vol. 42 (1936), pp. 544–546.

The remark is due, in substance, to Orrin Frink (Bulletin of the American Mathematical Society,vol. 34 (1928), pp. 329-333) that an (abstract) Boolean algebra can be represented as a ring in which every element is idempotent, the sum modulo 2 or symmetric difference—instead of the logical sum—being taken for the ring addition. Designating a ring in which every element is idempotent as a Boolean ring, the present author shows further that every Boolean ring with unit is, in the sense just described, a Boolean algebra; and that for every Boolean ring without unit there is a uniquely determined minimum Boolean ring with unit in which it can be embedded. These observations make it possible to deal with Boolean algebra from the point of view of the abstract theory of rings, as is done in the present paper (which is the first of a projected series of papers). The principal result is that every abstract system having the formal properties of a Boolean ring is isomorphic to an actual algebra of classes, the ring multiplication corresponding to logical multiplication of the classes, and the ring addition, to addition of the classes modulo 2. In order to obtain this result, use is made of the Zermelo hypothesis (axiom of choice) in the proof of the preliminary theorem that in a Boolean ring which contains at least two elements there exists at least one prime ideal. ALONZO CHURCH