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In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R, that is, a ring that consists of only idempotent elements.[1][2][3] 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 ∨,[4] 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.
- ^ Fraleigh 1976, pp. 25, 200
- ^ Herstein 1975, pp. 130, 268
- ^ McCoy 1968, p. 46
- ^ "Disjunction as sum operation in Boolean Ring".