An element p of a ring is idempotent if p2 = p. A Boolean ring is a ring with unit in which every element is idempotent. Warning: a ring with unit is by definition a ring with a distinguished element 1 that acts as a multiplicative identity and that is distinct from the additive identity 0. The effect of the last proviso is to exclude from consideration the trivial ring consisting of 0 alone. The phrase “with unit” is sometimes omitted from the definition of a Boolean ring; in that case our present concept is called a “Boolean ring with unit.”
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