Boolean ring
In
mathematics, a
Boolean ring R is a
ring for which
x2 =
x for all
x in
R; that is,
R consists of
idempotent elements. These rings arise from (and give rise to) Boolean algebras. One example is 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
- x + x = (x + x)2 = x2 + 2x2 + x2 = x + 2x + x
and we can subtract
x +
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).
If we define
- x &and y = xy,
- x ∨ y = x + y − xy,
- ~x = 1 + x
then these satisfy all of the axioms for meets, joins, and complements in a Boolean algebra. Thus every Boolean ring with 1 becomes a Boolean algebra. Similarly, every Boolean algebra becomes a Boolean ring with 1 thus:
- xy = x ∧ y,
- x + y = (x ∨ y) ∧ ~(x ∧ y).