Use this tag for questions related to Boolean rings such as the ring of integers modulo $2$ $\mathbb Z/2\mathbb Z$.
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
- $\mathbb Z/2\mathbb Z$ (AND is multiplication, XOR is addition)
- 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.
