  ## Encyclopedia > Boolean ring

Article Content

# 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).

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Great River, New York ... and 0.97% from two or more races. 1.88% of the population are Hispanic or Latino of any race. There are 509 households out of which 41.5% have children under the age of ...  