Boolean ring

In mathematics, a Boolean ring R is a ring (with identity) for which x = x for all x in R; that is, R consists only of idempotent elements. Boolean rings are automatically commutative and of characteristic 2 (see below for proof). A Boolean ring is essentially the same thing as a Boolean algebra, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨).... more

In mathematics, a Boolean ring R is a ring (with identity) for which x = x for all x in R; that is, R consists only of idempotent elements. Boolean rings are automatically commutative and of characteristic 2 (see below for proof). A Boolean ring is essentially the same thing as a Boolean algebra, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨). There are at least four different and incompatible systems of notation for Boolean rings and algebras. 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). 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... less