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A Boolean ring is a ring  that has a multiplicative identity, and in which every element is idempotent, that is,
 for all  Boolean rings are equivalent to Boolean algebras (or Boolean lattices). 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.
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