Boolean ring

In mathematics, a Boolean ring is a ring R such that x² = x for all x in R. These rings arise from (and give rise to) Boolean algebras, as is explained in that article.

Every Boolean ring R satisfies x + x = 0 for all x in R, because we know

1 + x = (1 + x)² = 1 + 2x + x² = 1 + 2x + x

and we can subtract 1 + x from both sides of this equation. A similar proof shows that every Boolean ring is commutative:

x + y = (x + y)² = x² + xy + yx + y² = x + xy + yx + y

and this yields xy + yx = 0, which means xy = -yx = yx (using the first property above).