Inmathematics, a Boolean ring R is aringfor which x2= x for all x in R,[1][2][3]such as the ring ofintegers modulo 2. That is, R consists only ofidempotent elements.[4][5]
Every Boolean ringgives riseto aBoolean algebra, with ring multiplication corresponding toconjunctionormeet∧, and ring addition toexclusive disjunctionorsymmetric difference(notdisjunction∨).
33.1 Notations
There are at least four different and incompatible systems of notation for Boolean rings and algebras.
• In commutative algebra the standard notation is to use x + y = (x ∧ ¬ y) ∨ (¬ x ∧ y) for the ring sum of x and y, and use xy = x ∧ y for their product.
• In logic, a common notation is to use x ∧ y for the meet (same as the ring product) and use x ∨ y for the join, given in terms of ring notation (given just above) by x + y + xy.
• In set theory and logic it is also common to use x · y for the meet, and x + y for the join x ∨ y. This use of + is different from the use in ring theory.
• A rare convention is to use xy for the product and x ⊕ y for the ring sum, in an effort to avoid the ambiguity of +.
The old terminology was to use “Boolean ring” to mean a “Boolean ring possibly without an identity”, and “Boolean algebra” to mean a Boolean ring with an identity. (This is the same as the old use of the terms “ring” and “algebra” in measure theory) (Also note that, when a Boolean ring has an identity, then a complement operation becomes definable on it, and a key characteristic of the modern definitions of both Boolean algebra andsigma-algebrais that they have complement operations.)
33.2 Examples
One example of a Boolean ring is thepower setof any set X, where the addition in the ring issymmetric difference, and the multiplication isintersection. As another example, we can also consider the set of allfiniteor cofinite subsets of X, again with symmetric difference and intersection as operations. More generally with these operations anyfield of setsis a Boolean ring. ByStone’s representation theoremevery Boolean ring is isomorphic to a field of sets (treated as a ring with these operations).
33.3 Relation to Boolean algebras
Since the join operation ∨ in a Boolean algebra is often written additively, it makes sense in this context to denote ring addition by ⊕, a symbol that is often used to denoteexclusive or.
88
x y x y x
x⋀y x⋁y ¬x
Venn diagramsfor the Boolean operations of conjunction, disjunction, and complement
Given a Boolean ring R, for x and y in R we can define x ∧ y = xy,
x ∨ y = x ⊕ y ⊕ xy,
¬x = 1 ⊕ x.
These operations then satisfy all of the axioms for meets, joins, and complements in aBoolean algebra. Thus every Boolean ring becomes a Boolean algebra. Similarly, every Boolean algebra becomes a Boolean ring thus:
xy = x ∧ y,
x ⊕ y = (x ∨ y) ∧ ¬(x ∧ y).
If a Boolean ring is translated into a Boolean algebra in this way, and then the Boolean algebra is translated into a ring, the result is the original ring. The analogous result holds beginning with a Boolean algebra.
A map between two Boolean rings is aring homomorphism if and only ifit is a homomorphism of the corresponding Boolean algebras. Furthermore, a subset of a Boolean ring is aring ideal(prime ring ideal, maximal ring ideal) if and only if it is anorder ideal(prime order ideal, maximal order ideal) of the Boolean algebra. Thequotient ringof a Boolean ring modulo a ring ideal corresponds to the factor algebra of the corresponding Boolean algebra modulo the corresponding order ideal.
33.4 Properties of Boolean rings
Every Boolean ring R satisfies x ⊕ x = 0 for all x in R, because we know x ⊕ x = (x ⊕ x)2= x2⊕ x2⊕ x2⊕ x2= x ⊕ x ⊕ x ⊕ x
and since <R,⊕> is an abelian group, we can subtract x ⊕ x from both sides of this equation, which gives x ⊕ x = 0.
A similar proof shows that every Boolean ring iscommutative:
x ⊕ y = (x ⊕ y)2= x2⊕ xy ⊕ yx ⊕ y2= x ⊕ xy ⊕ yx ⊕ y
and this yields xy ⊕ yx = 0, which means xy = yx (using the first property above).
The property x ⊕ x = 0 shows that any Boolean ring is anassociative algebraover thefieldF2with two elements, in just one way. In particular, any finite Boolean ring has ascardinalityapower of two. Not every associative algebra with one over F2is a Boolean ring: consider for instance thepolynomial ringF2[X].
The quotient ring R/I of any Boolean ring R modulo any ideal I is again a Boolean ring. Likewise, anysubringof a Boolean ring is a Boolean ring.
Everyprime idealP in a Boolean ring R ismaximal: thequotient ringR/P is anintegral domainand also a Boolean ring, so it is isomorphic to thefieldF2, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.
Boolean rings arevon Neumann regular rings.
Boolean rings are absolutely flat: this means that every module over them isflat.
Every finitely generated ideal of a Boolean ring isprincipal(indeed, (x,y)=(x+y+xy)).
Unification in Boolean rings is decidable,[6][7]that is, algorithms exist to solve arbitrary equations over Boolean rings.
33.5 Notes
[1] Fraleigh (1976, p. 200) [2] Herstein (1964, p. 91) [3] McCoy (1968, p. 46) [4] Fraleigh (1976, p. 25) [5] Herstein (1964, p. 224)
[6] Martin, U., Nipkow, T. (1986). “Unification in Boolean Rings”. In Jörg H. Siekmann. Proc. 8th CADE. LNCS 230.
Springer. pp. 506–513.
[7] A. Boudet,J.-P. Jouannaud, M. Schmidt-Schauß (1989). “Unification of Boolean Rings and Abelian Groups”(PDF).
Journal of Symbolic Computation 8: 449–477.doi:10.1016/s0747-7171(89)80054-9.
33.6 References
• Atiyah, Michael Francis;Macdonald, I. G.(1969), Introduction to Commutative Algebra, Westview Press,ISBN 978-0-201-40751-8
• Fraleigh, John B. (1976), A First Course In Abstract Algebra (2nd ed.), Reading: Addison-Wesley,ISBN 0-201-01984-1
• Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company,ISBN 978-1114541016
• McCoy, Neal H. (1968), Introduction To Modern Algebra (Revised ed.), Boston: Allyn and Bacon,LCCN 68015225
• Ryabukhin, Yu. M. (2001),“Boolean_ring”, in Hazewinkel, Michiel,Encyclopedia of Mathematics,Springer, ISBN 978-1-55608-010-4
33.7 External links
• John Armstrong,Boolean Rings