Rings and field, Every boolean ring is commutative.

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#booleanringiscommutative.
In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R, that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2.
Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨,which would constitute a semiring). Boolean rings are named after the founder of Boolean algebraA similar proof shows that every Boolean ring is commutative:
x + y = (x + y)2 = x2 + xy + yx + y2 = x + xy + yx + y and this yields xy + yx = 0, which means xy = yx (using the first property above).