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Concept
Boolean ring
Summary
In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R, that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2.
Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨, which would constitute a semiring). Conversely, every Boolean algebra gives rise to a Boolean ring. Boolean rings are named after the founder of Boolean algebra, George Boole.
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 +.
Historically, the term "Boolean ring" has been used to mean a "Boolean ring possibly without an identity", and "Boolean algebra" has been used to mean a Boolean ring with an identity. The existence of the identity is necessary to consider the ring as an algebra over the field of two elements: otherwise there cannot be a (unital) ring homomorphism of the field of two elements into the Boolean ring. (This is the same as the old use of the terms "ring" and "algebra" in measure theory.)
One example of a Boolean ring is the power set of any set X, where the addition in the ring is symmetric difference, and the multiplication is intersection. As another example, we can also consider the set of all finite or cofinite subsets of X, again with symmetric difference and intersection as operations.
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