A map between two Boolean rings is a ring homomorphism if and only if it is a homomorphism of the corresponding Boolean algebras.
12.
Another isomorphism of categories arises in the theory of Boolean algebras : the category of Boolean algebras is isomorphic to the category of Boolean rings.
13.
The quotient ring of a Boolean ring modulo a ring ideal corresponds to the factor algebra of the corresponding Boolean algebra modulo the corresponding order ideal.
14.
It can hence be shown ( by proving the distributive laws ) that the power set considered together with both of these operations forms a Boolean ring.
15.
The ring basis turns a Boolean algebra into a Boolean ring, namely a ring satisfying " x " 2 = " x ".
16.
Ideals in the set-theoretic sense are exactly ideals in the ring-theoretic sense on the Boolean ring formed by the powerset of the underlying set.
17.
19 ) " There is a one-to-one correspondence between Boolean algebras and Boolean rings and hence the name'Boolean'. " Reference ??
18.
Starting from this point, he soon focused his interest on the related theory of Boolean algebras and Boolean rings, and was thus led from logic to algebra.
19.
If a Boolean ring is translated into a Boolean algebra in this way, and then the Boolean algebra is translated into a ring, the result is the original ring.
20.
Handling this ambiguity was an early problem of the theory, reflecting the modern use of both Boolean rings and Boolean algebras ( which are simply different aspects of one type of structure ).
