Boolean ring

A Boolean ring is a ring in which every element satisfies x² = x. Its addition is necessarily commutative, and every element is its own additive inverse.

Ring (algebra): A set with addition and multiplication satisfying specified associativity, distributivity, and identity axioms. The Boolean ring condition strengthens the multiplication law of this broader algebraic structure.
Symmetric difference: The set of elements belonging to exactly one of two sets. Addition of sets in the power-set Boolean ring is symmetric difference.
Two-element field: The field with elements 0 and 1, with arithmetic modulo two. It is the simplest nonzero Boolean ring and the only Boolean ring that is a field.
Stone duality: A correspondence between Boolean algebras and compact Hausdorff totally disconnected spaces. Boolean rings encode Boolean algebras, making their structure accessible through topological spaces.
Boolean semiring: The two-element semiring in which addition is logical OR and multiplication is logical AND. It models Boolean logic but lacks additive inverses, unlike a Boolean ring.
Idempotent element: An element unchanged by applying a specified operation to it twice. Every element in a Boolean ring is idempotent under multiplication.
Intersection (set theory): The set of elements common to every set in a given collection. Multiplication in the power-set Boolean ring is set intersection.
Power set: The set of all subsets of a given set. With symmetric difference and intersection, a power set is a canonical Boolean ring.
Stone space: A compact Hausdorff totally disconnected topological space. Prime ideals of a Boolean ring form a Stone space that represents its algebraic structure.
Idempotent semiring: A semiring whose addition satisfies a + a = a for every element. Its idempotence concerns addition, whereas Boolean rings require idempotence of multiplication.

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