Boolean ring

Definition

A Boolean ring is a ring $R$ with identity such that $x^2 = x$ for every $x \in R$.

Remarks

In a Boolean ring, $x + x = 0$ for every $x$, and the ring is commutative. The ring $\mathbb{Z}/2\mathbb{Z}$ is a Boolean ring. The power set of a set is a Boolean ring, with symmetric difference as addition and intersection as multiplication. Each Boolean ring is a Boolean algebra with $x \wedge y = xy$, $x \vee y = x + y + xy$ and $\neg x = 1 + x$. This gives a correspondence between Boolean rings and Boolean algebras.

Explanations

L1 human

A Boolean ring is a ring in which squaring changes nothing. It is the same thing as a Boolean algebra, written with plus and times instead of or and and.