On the other hand there is a sense in which Boolean algebras and Boolean rings can be viewed as subvarieties of each other even though they have different signatures, because of the translation between them allowing every Boolean algebra to be understood as a Boolean ring and conversely; in this sort of situation the homomorphisms between corresponding structures are the same.
32.
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 . ( This is the same as the old use of the terms " ring " and " algebra " in measure theory .)
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