Editing Boolean ring (section)

{{Short description|Algebraic structure in mathematics}}
In [[mathematics]], a '''Boolean ring''' {{math|''R''}} is a [[ring (mathematics)|ring]] for which {{math|1=''x''<sup>2</sup> = ''x''}} for all {{math|''x''}} in {{math|''R''}}, that is, a ring that consists of only [[idempotent element (ring theory)|idempotent element]]s.{{sfn|Fraleigh|1976|pp=25,200|ps=none}}{{sfn|Herstein|1975|pp=130,268|ps=none}}{{sfn|McCoy|1968|p=46|ps=none}} An example is the ring of [[modular arithmetic#Integers modulo m|integers modulo 2]].

Every Boolean ring gives rise to a [[Boolean algebra (structure)|Boolean algebra]], with ring multiplication corresponding to [[logical conjunction|conjunction]] or [[meet (mathematics)|meet]] {{math|∧}}, and ring addition to [[exclusive or|exclusive disjunction]] or [[symmetric difference]] (not [[logical disjunction|disjunction]] {{math|∨}},{{refn|{{cite web|url=https://math.stackexchange.com/q/1621618|title = Disjunction as sum operation in Boolean Ring}}}} which would constitute a [[semiring]]).  Conversely, every Boolean algebra gives rise to a Boolean ring.  Boolean rings are named after the founder of Boolean algebra, [[George Boole]].