The theory of representations for Boolean algebras
M. H. Stone · Transactions of the American Mathematical Society · 1936
ring B which possesses a unit element, in such a manner that B is unique in the following sense : if C is a Boolean ring with unit containing A, then C contains also a Boolean ring B* isomorphic to B and containing A. A finite Boolean ring necessarily possesses a unit and has a cardinal number which is a power of 2.In this, as in all subsequent discussions, we may use the familiar rules governing the ring operations without going into complete detail.Using such rules, we see that in a Boolean ring a + b = (a + b)(a + b) = (a + b) + (ba + ab) and hence that ba+ab = 0.If we put b = a in the latter relation, we find at once that a+a = 0, or, equivalently, a= -a.Using this result, we conclude that ba= -iab) = ab, thus establishing the commutative law for multiplication.The special rules which we have now demonstrated will henceforth be used in our discussions without explicit reference.In a Boolean ring with more than two elements, we can choose a and b so that a¿¿0, b^O, a^b.If ab = 0, then a and b are both divisors of 0. On the other hand, if ab^O, then ab and a+b are both divisors of 0: for a+b = 0 would imply a= -b = b, contrary to hypothesis; and abia+b) =aab+abb = ab+ab = 0.A Boolean ring with one or with two elements obviously cannot contain divisors of 0, every product in such a ring either containing 0 as a factor or reducing, by the law of idempotence, to an element other than 0.In discussing the possibility of imbedding a Boolean ring A in a Boolean ring B with unit, we may disregard the trivial case where A has a unit and B coincides with A. When A has no unit, we construct B by the adjunction of suitable elements.The construction can be carried out even when A has a unit and always produces B as a proper superclass of A. We first provide an abstract element e, distinct from those of A, and define tt = t, ta = at = a, t + 0 = 0 + t = t, e + e = 0, observing that the elements 0 and e constitute a two-element Boolean ring.We then consider the ordered pairs (a, a) where a is in A and a = 0 or a = e, defining the operations of addition and multiplication upon them by the rules ia,a) + ib,ß) = ia + b,a + ß), (a, a)ib, ß) = iab + ab + aß, aß).It is easily verified that under these operations the class of pairs (a, a) is a Boolean ring with (0, e) as its unit.Instead of giving all the calculations in detail, we shall discuss only one or two steps.Passing over the commutative