 Boolean ring

A Boolean ring is a unital ring satisfying $a^2=a$ for every element. Expanding $(a+1{)}^2=a+1$ gives $2a=0$ for every $a$. A nonzero Boolean ring therefore has characteristic two. Expanding $(a+b{)}^2=a+b$ then gives $ab+ba=0$, which in characteristic two says $ab=ba$: Boolean rings are automatically commutative.

The zero ring is Boolean too, but has characteristic one under the least-positive-integer convention used here. Products of copies of ${\mathbb{F}}_2$, such as ${\mathbb{F}}_2\times {\mathbb{F}}_2$, give nonzero examples; squaring acts coordinatewise as the identity. Every prime ideal of a Boolean ring is maximal because its integral-domain quotient has only zero and one.

A **Boolean ring** is a specific type of ring in the field of abstract algebra that has particular properties.