Boolean ring

In

• In commutative algebra the standard notation is to use x + y = (x ∧ ¬ y) ∨ (¬ x ∧ y) for the ring sum of x and y, and use xy = x ∧ y for their product.
• In logic, a common notation is to use x ∧ y for the meet (same as the ring product) and use x ∨ y for the join, given in terms of ring notation (given just above) by x + y + xy.
• In set theory and logic it is also common to use x · y for the meet, and x + y for the join x ∨ y.^[5] This use of + is different from the use in ring theory.
• A rare convention is to use xy for the product and x ⊕ y for the ring sum, in an effort to avoid the ambiguity of +.

Historically, the term "Boolean ring" has been used to mean a "Boolean ring possibly without an identity", and "Boolean algebra" has been used to mean a Boolean ring with an identity. The existence of the identity is necessary to consider the ring as an algebra over the

One example of a Boolean ring is the power set of any set X, where the addition in the ring is symmetric difference, and the multiplication is intersection. 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. More generally with these operations any field of sets is a Boolean ring. By Stone's representation theorem every Boolean ring is isomorphic to a field of sets (treated as a ring with these operations).

Relation to Boolean algebras

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.

Given a Boolean ring R, for x and y in R we can define

x ∧ y = xy,

x ∨ y = x ⊕ y ⊕ xy,

¬x = 1 ⊕ x.

These operations then satisfy all of the axioms for meets, joins, and complements in a Boolean algebra. Thus every Boolean ring becomes a Boolean algebra. Similarly, every Boolean algebra becomes a Boolean ring thus:

xy = x ∧ y,

x ⊕ y = (x ∨ y) ∧ ¬(x ∧ y).

If a Boolean ring is translated into a Boolean algebra in this way, and then the Boolean algebra is translated into a ring, the result is the original ring. The analogous result holds beginning with a Boolean algebra.

A map between two Boolean rings is a

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. Not every unital associative algebra over F₂ is a Boolean ring: consider for instance the polynomial ring F₂[X].

The quotient ring R / I of any Boolean ring R modulo any ideal I is again a Boolean ring. Likewise, any subring of a Boolean ring is a Boolean ring.

Any

The maximal ring of quotients Q(R) (in the sense of Utumi and Lambek) of a Boolean ring R is a Boolean ring, since every partial endomorphism is idempotent.^[6]

Every prime ideal P in a Boolean ring R is maximal: the quotient ring R / P is an integral domain and also a Boolean ring, so it is isomorphic to the field F₂, which shows the maximality of P. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.

Every finitely generated ideal of a Boolean ring is principal (indeed, (x,y) = (x + y + xy)). Furthermore, as all elements are idempotents, Boolean rings are commutative von Neumann regular rings and hence absolutely flat, which means that every module over them is flat.

Unification