Boolean ring
A Boolean ring is a ring that has a multiplicative identity
,
and in which every element is idempotent
, that is,
Boolean rings are necessarily commutative (http://planetmath.org/CommutativeRing).
Also, if is a Boolean ring, then for each .
Boolean rings are equivalent to Boolean algebras
(or Boolean lattices (http://planetmath.org/BooleanLattice)).
Given a Boolean ring ,
define and and
for all ,
then is a Boolean algebra.
Given a Boolean algebra ,
define and ,
then 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 is any set, then the power set forms a Boolean ring, with intersection
as multiplication and symmetric difference
as addition.
Let be the ring with the operations being coordinate-wise.
Then we can check:
the four elements that form the ring are idempotent. So is Boolean.
Title | Boolean ring |
---|---|
Canonical name | BooleanRing |
Date of creation | 2013-03-22 12:27:28 |
Last modified on | 2013-03-22 12:27:28 |
Owner | yark (2760) |
Last modified by | yark (2760) |
Numerical id | 24 |
Author | yark (2760) |
Entry type | Definition |
Classification | msc 06E99 |
Classification | msc 03G05 |
Related topic | Idempotency |
Related topic | BooleanLattice |
Related topic | BooleanIdeal |