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.
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.
|Date of creation||2013-03-22 12:27:28|
|Last modified on||2013-03-22 12:27:28|
|Last modified by||yark (2760)|