Boolean ringIn mathematics, a Boolean ring is a ring R such that x2 = x for all x in R; that is, R consists of idempotent elements. These rings arise from (and give rise to) Boolean algebras. Examples are given by the power set of any set X, where the addition in the ring is symmetric difference, and the multiplication is intersection.
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
- x + y = (x + y)2 = x2 + xy + yx + y2 = x + xy + yx + y