## Encyclopedia > Boolean ring

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# 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).

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