A ring with unit is Boolean if the operation of multiplication is idempotent; that is, for every element . Although the terminology would make sense for rings without unit, the common usage assumes a unit.
Boolean rings and the ring homomorphisms between them form a category .
- has characteristic (meaning that for all ):
- is commutative (meaning that for all ):
Define to mean . Then:
Next define to be . Then:
- is a pseudocomplement of (meaning that ):
By relativising from to , we can show that is a Heyting algebra.
- But don't bother, because is also an op-pseudocomplement of :
Therefore, is a complement of , and is a Boolean algebra.
Conversely, starting with a Boolean algebra (with the meet written multiplicatively), let be (which is called exclusive disjunction in and symmetric difference in ). Then is a Boolean ring.
In fact, we have:
Boolean rings and Boolean algebras are equivalent.
This extends to an equivalence of concrete categories; that is, given the underlying set , the set of Boolean ring structures on is naturally (in ) bijective with the set of Boolean algebra structures on .
Here is a very convenient result: although a Boolean ring is a rig in two different ways (as a ring or as a distributive lattice), these have the same concept of ideal!
The most common example is the power set of any set . It is a Boolean ring with symmetric difference as the addition and the intersection of sets as the multiplication.
Back in the day, the term ‘ring’ meant (more often than now is the case) a possibly nonunital ring; that is a semigroup, rather than a monoid, in Ab. This terminology applied also to Boolean rings, and it changed even more slowly. Thus older books will make a distinction between ‘Boolean ring’ (meaning an idempotent semigroup in ) and ‘Boolean algebra’ (meaning an idempotent monoid in ), in addition to (or even instead of) the difference between and as fundamental operation. This distinction survives most in the terminology of -rings and -algebras.
Inasmuch as a semilattice is a commutative idempotent monoid, a Boolean ring may be defined as a semilattice in . However, with Boolean rings, we do not need to hypothesize commutativity; it follows.