Boolean ring

As another example, we can also consider the set of all finite or cofinite subsets of X, again with symmetric difference and intersection as operations.

Since the join operation ∨ in a Boolean algebra is often written additively, it makes sense in this context to denote ring addition by ⊕, a symbol that is often used to denote exclusive or.

Every Boolean ring R satisfies x ⊕ x = 0 for all x in R, because we know and since (R, ⊕) is an abelian group, we can subtract x ⊕ x from both sides of this equation, which gives x ⊕ x = 0.

A similar proof shows that every Boolean ring is commutative: The property x ⊕ x = 0 shows that any Boolean ring is an associative algebra over the field F2 with two elements, in precisely one way.

[citation needed] In particular, any finite Boolean ring has as cardinality a power of two.

Every finitely generated ideal of a Boolean ring is principal (indeed, (x,y) = (x + y + xy)).

Unification in Boolean rings is unitary if all the uninterpreted function symbols are nullary and finitary otherwise (i.e. if the function symbols not occurring in the signature of Boolean rings are all constants then there exists a most general unifier, and otherwise the minimal complete set of unifiers is finite).