Boolean ring
Noun
Meaning
Boolean ring (plural Boolean rings)
(algebra) A ring whose multiplicative operation is idempotent.
From the defining idempotency property of a Boolean ring it is possible to prove that such ring has to have the further properties that each element is its own inverse and that such ring must be commutative.
Examples
By Stone's representation theorem every Boolean ring is isomorphic to a field of sets (treated as a ring with these operations).
Furthermore, a subset of a Boolean ring is a ring ideal (prime ring ideal, maximal ring ideal) if and only if it is an order ideal (prime order ideal, maximal order ideal) of the Boolean algebra.
Historically, the term "Boolean ring" has been used to mean a "Boolean ring possibly without an identity", and "Boolean algebra" has been used to mean a Boolean ring with an identity.
More generally with these operations any field of sets is a Boolean ring.
Moreover, these notions coincide with ring theoretic ones of prime ideal and maximal ideal in the Boolean ring A. The dual of an ideal is a filter.
The existence of the identity is necessary to consider the ring as an algebra over the field of two elements : otherwise there cannot be a (unital) ring homomorphism of the field of two elements into the Boolean ring.