Postulate-sets for Boolean Rings

1. Introduction. Boolean rings (or generalized Boolean algebras), defined by Stone, are rings in which every element is idempotent. Sets of postulates for these rings have been given by Stone(2) and by Stabler(8). I give in this paper a number of additional postulate-sets for such rings. The sets are all expressed in terms of ring addition and multiplication. Stone's postulates for Boolean rings in terms of ring operations are given as set S below; Stabler's postulates of this type are sets S' and S" below. The additional sets of the present paper are sets I-IX below. Stone has not undertaken the task of eliminating possible redundancies from his set S, and Stabler raised the question of independence of his set S' when a unit-element postulate is added to S'. It will be shown that, except for an overlooked redundancy, set S is independent and that S' remains independent after the addition of a unit-element postulate. The latter fact will be shown to hold also for set S, without the redundancy, and for sets S",