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Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x ∨ y := x + y + ( x · y ) and x ∧ y := x · y.
Since these two constructions are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa.
Furthermore, a map f: A → B is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings.
The categories of Boolean rings and Boolean algebras are equivalent.

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