Editing Boolean ring (section)

== Properties of Boolean rings ==
Every Boolean ring {{math|''R''}} satisfies {{math|1=''x'' ⊕ ''x'' = 0}} for all {{math|''x''}} in {{math|''R''}}, because we know
: {{math|1=''x'' ⊕ ''x'' = (''x'' ⊕ ''x'')<sup>2</sup> = ''x''<sup>2</sup> ⊕ ''x''<sup>2</sup> ⊕ ''x''<sup>2</sup> ⊕ ''x''<sup>2</sup> = ''x'' ⊕ ''x'' ⊕ ''x'' ⊕ ''x''}}
and since {{math|(''R'', ⊕)}} is an abelian group, we can subtract {{math|''x'' ⊕ ''x''}} from both sides of this equation, which gives {{math|1=''x'' ⊕ ''x'' = 0}}. A similar proof shows that every Boolean ring is [[commutative]]:
: {{math|1=''x'' ⊕ ''y'' = (''x'' ⊕ ''y'')<sup>2</sup> = ''x''<sup>2</sup> ⊕ ''xy'' ⊕ ''yx'' ⊕ ''y''<sup>2</sup> = ''x'' ⊕ ''xy'' ⊕ ''yx'' ⊕ ''y''}} and this yields {{math|1=''xy'' ⊕ ''yx'' = 0}}, which means {{math|1=''xy'' = ''yx''}} (using the first property above).

The property {{math|1=''x'' ⊕ ''x'' = 0}} shows that any Boolean ring is an [[associative algebra]] over the [[field (mathematics)|field]] {{math|'''F'''<sub>2</sub>}} with two elements, in precisely one way.{{citation needed|date=March 2023}} In particular, any finite Boolean ring has as [[cardinality]] a [[power of two]]. Not every unital associative algebra over {{math|'''F'''<sub>2</sub>}} is a Boolean ring: consider for instance the [[polynomial ring]] {{math|'''F'''<sub>2</sub>[''X'']}}.

The quotient ring {{math|1=''R'' / ''I''}} of any Boolean ring {{math|''R''}} modulo any ideal {{math|''I''}} is again a Boolean ring. Likewise, any [[subring]] of a Boolean ring is a Boolean ring.

Any [[localization_of_a_ring|localization]] {{math|''RS''<sup>−1</sup>}} of a Boolean ring {{math|''R''}} by a set {{math|''S'' ⊆ ''R''}} is a Boolean ring, since every element in the localization is idempotent.

The maximal ring of quotients {{math|''Q''(''R'')}} (in the sense of Utumi and [[Joachim Lambek|Lambek]]) of a Boolean ring {{math|''R''}} is a Boolean ring, since every partial endomorphism is idempotent.{{sfn|Brainerd|Lambek|1959|loc=Corollary 2|ps=none}}

Every [[prime ideal]] {{math|''P''}} in a Boolean ring {{math|''R''}} is [[maximal ideal|maximal]]: the [[quotient ring]] {{math|''R'' / ''P''}} is an [[integral domain]] and also a Boolean ring, so it is isomorphic to the [[field (mathematics)|field]] {{math|'''F'''<sub>2</sub>}}, which shows the maximality of {{math|''P''}}. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.

Every finitely generated ideal of a Boolean ring is [[principal ideal|principal]] (indeed, {{math|1=(''x'',''y'') = (''x'' + ''y'' + ''xy''))}}. Furthermore, as all elements are idempotents, Boolean rings are commutative [[von Neumann regular ring]]s and hence absolutely flat, which means that every module over them is [[flat module|flat]].