Editing Boolean ring (section)

== Relation to Boolean algebras ==
[[File:Vennandornot.svg|center|500px|thumb|[[Venn diagram]]s for the Boolean operations of conjunction, disjunction, and complement]]
Since the join operation {{math|∨}} in a Boolean algebra is often written additively, it makes sense in this context to denote ring addition by {{math|⊕}}, a symbol that is often used to denote [[exclusive or]].

Given a Boolean ring {{math|''R''}}, for {{math|''x''}} and {{math|''y''}} in {{math|''R''}} we can define
: {{math|1=''x'' ∧ ''y'' = ''xy''}},
: {{math|1=''x'' ∨ ''y'' = ''x'' ⊕ ''y'' ⊕ ''xy''}},
: {{math|1=¬''x'' = 1 ⊕ ''x''}}.

These operations then satisfy all of the axioms for meets, joins, and complements in a [[Boolean algebra (structure)|Boolean algebra]]. Thus every Boolean ring becomes a Boolean algebra.  Similarly, every Boolean algebra becomes a Boolean ring thus:
: {{math|1=''xy'' = ''x'' ∧ ''y'',}}
: {{math|1=''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.