Tag Info

Use this tag for questions related to Boolean rings such as the ring of integers modulo 2.

A Boolean ring R is a ring for which x$^2$ = x for all x in R. That is, R consists of idempotent elements only.

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).

Examples of Boolean rings include

• the ring of integers modulo 2,
• the power set of any set X where addition is symmetric difference, and multiplication is intersection, and
• the set of all finite or co-finite subsets of X, again with symmetric difference and intersection as operations.

More generally, with the operations of the last two examples, any field of sets is a Boolean ring. By Stone's representation theorem, every Boolean ring is isomorphic to a field of sets treated as a ring with those operations.