In
mathematics, a
Boolean ring is a
ring R such that
x^{2} =
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 + x^{2} = 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} = x^{2} + xy + yx + y^{2} = x + xy + yx + y
and this yields
xy +
yx = 0, which means
xy = -
yx =
yx (using the first property above).
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