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Boolean ring

An associative ring with identity element is called a Boolean ring if all its elements are idempotent, that is   

(1)

Boolean rings have many interesting properties. For instance,  multiplication of a Boolean ring is commutative, and every element is its own additive inverse, that is, we have

Theorem. Multiplication of a Boolean ring is commutative and satisfies the identity

(2)

Proof. Let , then

This implies that

(3)

Taking in  (3)  and using  (1) we get  (2) .  Knowing that relation  (3)  yields the commutativity. QED          

Theorem. The cardinality of  any finite Boolean ring is a power of two.

Proof. Equation (2)  says that any Boolean ring is an associative algebra over the field with two elements. QED

There immediately follows from the definition that any subring of a Boolean ring is again a Boolean ring. Consequently, the quotient ring    of a Boolean ring modulo an ideal is again a Boolean ring.

Theorem. Every prime ideal in a Boolean ring is maximal.

Proof.  The quotient ring R/P is an integral domain and simultaneously a Boolean ring, so it must be isomorphic to the field . This implies the maximality of . QED

Maximal ideals in a ring with identity are always prime, thus we conclude that:

Corollary. Prime ideals and maximal ideals coincide in a Boolean ring.

One important example of a Boolean ring is the power set of any set , where the addition in the ring is symmetric difference, and the multiplication is intersection.

As another example, we can also consider the set of all finite subsets of  , again with respect to the operations of symmetric difference and intersection. More generally,  any field of sets with respect to these operations is a Boolean ring. By the Stone's theorem   every Boolean ring is isomorphic to a field of sets (if considered as a ring with respect to these operations).

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