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Boolean ring∗
yark†
2013-03-21 13:43:21
A Boolean ring is a ring R that has a multiplicative identity, and in which
every element is idempotent, that is,
x2 = x for all x ∈ R.
Boolean rings are necessarily commutative. Also, if R is a Boolean ring, then
x = −x for each x ∈ R.
Boolean rings are equivalent to Boolean algebras (or Boolean lattices). Given
a Boolean ring R, define x ∧ y = xy and x ∨ y = x + y + xy and x0 = x + 1 for all
x, y ∈ R, then (R, ∧, ∨, 0 , 0, 1) is a Boolean algebra. Given a Boolean algebra
(L, ∧, ∨, 0 , 0, 1), define x · y = x ∧ y and x + y = (x0 ∧ y) ∨ (x ∧ y 0 ), then (L, ·, +)
is a Boolean ring. In particular, the category of Boolean rings is isomorphic to
the category of Boolean lattices.
Examples
As mentioned above, every Boolean algebra can be considered as a Boolean ring.
In particular, if X is any set, then the power set P(X) forms a Boolean ring,
with intersection as multiplication and symmetric difference as addition.
Let R be the ring Z2 × Z2 with the operations being coordinate-wise. Then
we can check:
(1, 1) × (1, 1)
=
(1, 1)
(1, 0) × (1, 0)
=
(1, 0)
(0, 1) × (0, 1)
=
(0, 1)
(0, 0) × (0, 0)
=
(0, 0)
the four elements that form the ring are idempotent. So R is Boolean.
∗ hBooleanRingi
created: h2013-03-21i by: hyarki version: h32602i Privacy setting: h1i
hDefinitioni h06E99i h03G05i
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