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system with all ten properties (property (7) follows from (l)-(3))we consider it as a system with double composition in which the fundamental operations are multiplication and an operation + defined by the equation a + b = (a Ab) Aab. If we introduce also the operation ' defined by the equation a' = a Ae, we easily deduce the following properties: (a)a+b = b+a; (G )a+(b+c) = (a+b) +c; (7y)ab = (a' + b')'; (5) ab + ab' = a. These properties show that R is a Boolean algebra in accordance with a recent paper of Huntington;4 and the elements 0 and e in R are easily identified as the Boolean zero and unit, respectively. In the remainder of the note, we revert to the usual notation for ring addition, writing + in place of A in (l)-(10); and we shall drop the requirement of the existence of a unit. Thus we shall consider an abstract commutative ring R with the special properties (8) and (9), (8) now being written as a + a = 0, or, equivalently, a = -a. We omit (7) as a consequence of (l)-(3). We may also, according to a suggestion of J. v Neumann, omit (8) as a consequence of the other properties. Thus we consider a commutative ring all of whose elements are idempotent. By specializing to such a ring the well-known theorems concerning the homomorphisms of rings,3 we can immediately obtain Theorem I of our previous communication.1 We now show that Krull's theory of ideal arithmetic in rings with absolute values6 is applicable to R, at least in principle. To this end, we determine all real-valued functions a I with argument a in R which assume two or more distinct values and which have the property (11) abl = a bl. By (9) and (11), it is clear that a = a 2 and hence that the value of a I is either 0 or 1. Using the relation 0 = Oa and choosing a so that IaI =O,wefindthatIO II = Oa|I= IO IaIO = . Since(a+b)ab = 0, it is clear that a + b I I a I I b J = I 0 | = 0 and hence that the relations la l = I b = 1 imply a + b I = 0. By a simple combinatorial analysis and the relation b = -b, we now find that (12) Ia + b . max ( a |, lb I); (13) Ia -b = Oifandonlyif lal = b j. The relations (11) and (12) are precisely those upon which Krull bases his theory. From (13) we can show without difficulty that the classes 2I and e8 specified by the respective relations I a I = 0, I a = 1 have the following properties: Kt is an ideal in R; 2l and e are both residual classes (mod 21) and they are the only such classes; and hence the ideal 21 is both prime and divisorless. Every ideal 2 with these properties determines an absolute value I a in R. Once the existence of prime ideals in R is established, Krull's results serve to establish Theorem III of our previous communica104 PROC. N. A. S. MATHEMA TICS: M. H. STONE tion; and they yield also the first part of Theorem IV1, up to the words "with the following properties." We have thus shown that the theory of Boolean algebras, with particular reference to ideal-theoretic and arithmetic properties, is a special case of the theory of rings. We can say further that the theory of those algebraic systems known as distributive lattices' can be considered as a part of the theory of rings: for an unpublished result due to Mr. Holbrook MacNeille shows that every distributive lattice can be imbedded by a purely algebraic construction in a Boolean algebra. An interesting consequence of the theory developed in this note is that it is now possible to describe the mathematical theory of probability entirely in the language of the theory of rings. It is the theory7 of functions p(a) which are defined over a ring R of the special type described above, which assume values in an ordered ring P, and which have the special properties p(a) > 0, p(a + b) = p(a) + p(b) when ab = 0. The elements of R are events, and p(a) is the relative probability of the event a. In general, it is not necessary to assume that P is a commutative ring or that the ordering in P is archimedean. If we assume that P is a commutative ring with archimedean order, then P is isomorphic to a subring of the real number system. We point out that the construction of the absolute value a discussed above provides a simple illustration of the present theory, since la I 2 Oandsinceab = Oimplies a + b I = a I+ b I. 1 Proc. Nat. Acad. Sci., 20, 197-202 (1934). 2 All of these properties have been known for some time. See, for example, Daniell, Bull. Am. Math. Soc., 23, 446-458(1916); B. A. Bernstein, Trans. Am. Math. Soc., 26, 171-175 (1924), and 28, 654-657 (1926). 3See B. L. van der Waerden, Moderne Algebra, 1, Chapter 3 (Berlin, 1930). 4Huntington, Trans. Am. Math. Soc., 35, 274-304 and 557-558, especially the latter, (1933). 5 See Ore, Bull. Am. Math. Soc., 39, 728-745 (1934), especially 742-745; and Krull, Math. Zeit., 29, 42-54 (1929), Math. Zeit., 31, 526-557 (1930), and J. fur Math., 167, 157-196 (1932). 6 See Garrett Birkhoff, Proc. Camb. Phil. Soc., 29, 441-464, especially 442 and 453 (1933). 7 See Kolmogoroff, Grundbegrife der Wahrscheinlichkeitsrechnung, 2 (Berlin, 1933), and B. L. van der Waerden, Moderne Algebra, 1, Chapter 10, the latter for the theory of ordered rings and fields (Berlin, 1930). VOL. 21, 1935 105

Subsumption of the Theory of Boolean Algebras under the Theory of Rings