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 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!
Back in the day, the term ‘ring’ meant 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 hypothesise commutativity; it follows.
Revised on February 17, 2012 23:43:49
by Anonymous Hero