Boolean Ring - Relation To Boolean Algebras

Relation To Boolean Algebras

Since the join operation ∨ in a Boolean algebra is often written additively, it makes sense in this context to denote ring addition by ⊕, a symbol that is often used to denote exclusive or.

Given a Boolean ring R, for x and y in R we can define

x ∧ y = xy,

x ∨ y = x ⊕ y ⊕ xy,

¬x = 1 ⊕ x.

These operations then satisfy all of the axioms for meets, joins, and complements in a Boolean algebra. Thus every Boolean ring becomes a Boolean algebra. Similarly, every Boolean algebra becomes a Boolean ring thus:

xy = x ∧ y,

x ⊕ y = (x ∨ y) ∧ ¬(x ∧ y).

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. The analogous result holds beginning with a Boolean algebra.

A map between two Boolean rings is a ring homomorphism if and only if it is a homomorphism of the corresponding Boolean algebras. 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. 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.