Boolean ring

Encyclopedia

In mathematics

, a

(with identity) for which

A Boolean ring is almost the same thing as a Boolean algebra, with ring multiplication corresponding to conjunction

or meet

∧, and ring addition to exclusive disjunction or symmetric difference

(not disjunction

∨), except for the fact that the absorption law

is violated.

The old terminology was to use "Boolean ring" to mean a "Boolean ring possibly without an identity", and "Boolean algebra" to mean a Boolean ring with an identity. (This is the same as the old use of the terms "ring" and "algebra" in measure theory) (Also note that, when a Boolean ring has an identity, then a complement operation becomes definable on it, and a key characteristic of the modern definitions of both Boolean algebra and sigma-algebra

is that they have complement operations.)

, and the multiplication is intersection

. As another example, we can also consider the set of all finite or cofinite subsets of

every Boolean ring is isomorphic to a field of sets (treated as a ring with these operations).

Given a Boolean ring

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:

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 ring homomorphism

if and only if

it is a homomorphism of the corresponding Boolean algebras. Furthermore, a subset of a Boolean ring is a ring ideal (prime ring ideal, maximal ring ideal) if and only if it is an order ideal (prime order ideal, maximal order ideal) of the Boolean algebra. The quotient ring

of a Boolean ring modulo a ring ideal corresponds to the factor algebra of the corresponding Boolean algebra modulo the corresponding order ideal.

and since <

and this yields

The property

over the field

. Not every associative algebra with one over

The quotient ring

of a Boolean ring is a Boolean ring.

Every prime ideal

: the quotient ring

Boolean rings are von Neumann regular ring

s.

Boolean rings are absolutely flat: this means that every module over them is flat

.

Every finitely generated ideal of a Boolean ring is principal

(indeed,

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a

**Boolean ring***R*is a ringRing (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

(with identity) for which

*x*^{2}=*x*for all*x*in*R*; that is,*R*consists only of idempotent elements.A Boolean ring is almost the same thing as a Boolean algebra, with ring multiplication corresponding to conjunction

Logical conjunction

In logic and mathematics, a two-place logical operator and, also known as logical conjunction, results in true if both of its operands are true, otherwise the value of false....

or meet

Meet (mathematics)

In mathematics, join and meet are dual binary operations on the elements of a partially ordered set. A join on a set is defined as the supremum with respect to a partial order on the set, provided a supremum exists...

∧, and ring addition to exclusive disjunction or symmetric difference

Symmetric difference

In mathematics, the symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by A\,\Delta\,B\,orA \ominus B....

(not disjunction

Logical disjunction

In logic and mathematics, a two-place logical connective or, is a logical disjunction, also known as inclusive disjunction or alternation, that results in true whenever one or more of its operands are true. E.g. in this context, "A or B" is true if A is true, or if B is true, or if both A and B are...

∨), except for the fact that the absorption law

Absorption law

In algebra, the absorption law or absorption identity is an identity linking a pair of binary operations.Two binary operations, say ¤ and *, are said to be connected by the absorption law if:...

is violated.

## Notations

There are at least four different and incompatible systems of notation for Boolean rings and algebras.- 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*. 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 +.

The old terminology was to use "Boolean ring" to mean a "Boolean ring possibly without an identity", and "Boolean algebra" to mean a Boolean ring with an identity. (This is the same as the old use of the terms "ring" and "algebra" in measure theory) (Also note that, when a Boolean ring has an identity, then a complement operation becomes definable on it, and a key characteristic of the modern definitions of both Boolean algebra and sigma-algebra

Sigma-algebra

In mathematics, a σ-algebra is a technical concept for a collection of sets satisfying certain properties. The main use of σ-algebras is in the definition of measures; specifically, the collection of sets over which a measure is defined is a σ-algebra...

is that they have complement operations.)

## Examples

One example of a Boolean ring is the power set of any set*X*, where the addition in the ring is symmetric differenceSymmetric difference

In mathematics, the symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by A\,\Delta\,B\,orA \ominus B....

, and the multiplication is intersection

Intersection (set theory)

In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B , but no other elements....

. 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 theoremStone's representation theorem for Boolean algebras

In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Stone...

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 ring homomorphism

Ring homomorphism

In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....

if and only if

If and only if

In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....

it is a homomorphism of the corresponding Boolean algebras. Furthermore, a subset of a Boolean ring is a ring ideal (prime ring ideal, maximal ring ideal) if and only if it is an order ideal (prime order ideal, maximal order ideal) of the Boolean algebra. The quotient ring

Quotient ring

In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...

of a Boolean ring modulo a ring ideal corresponds to the factor algebra of the corresponding Boolean algebra modulo the corresponding order ideal.

## Properties of Boolean rings

Every Boolean ring*R*satisfies*x*⊕*x*= 0 for all*x*in*R*, because we know*x*⊕*x*= (*x*⊕*x*)^{2}=*x*^{2}⊕ 2*x*^{2}⊕*x*^{2}=*x*⊕ 2*x*⊕*x*=*x*⊕*x*⊕*x*⊕*x*

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:*x*⊕*y*= (*x*⊕*y*)^{2}=*x*^{2}⊕*xy*⊕*yx*⊕*y*^{2}=*x*⊕*xy*⊕*yx*⊕*y*

and this yields

*xy*⊕*yx*= 0, which means*xy*=*yx*(using the first property above).The property

*x*⊕*x*= 0 shows that any Boolean ring is an associative algebraAssociative algebra

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...

over the field

Field (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

**F**_{2}with two elements, in just one way. In particular, any finite Boolean ring has as cardinality a power of twoPower of two

In mathematics, a power of two means a number of the form 2n where n is an integer, i.e. the result of exponentiation with as base the number two and as exponent the integer n....

. Not every associative algebra with one over

**F**_{2}is a Boolean ring: consider for instance the polynomial ringPolynomial ring

In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...

**F**_{2}[*X*].The quotient ring

*R*/*I*of any Boolean ring*R*modulo any ideal*I*is again a Boolean ring. Likewise, any subringSubring

In mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...

of a Boolean ring is a Boolean ring.

Every prime ideal

Prime ideal

In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...

*P*in a Boolean ring*R*is maximalMaximal ideal

In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...

: the quotient ring

Quotient ring

In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...

*R*/*P*is an integral domain and also a Boolean ring, so it is isomorphic to the fieldField (mathematics)

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...

**F**_{2}, which shows the maximality of*P*. Since maximal ideals are always prime, prime ideals and maximal ideals coincide in Boolean rings.Boolean rings are von Neumann regular ring

Von Neumann regular ring

In mathematics, a von Neumann regular ring is a ring R such that for every a in R there exists an x in R withOne may think of x as a "weak inverse" of a...

s.

Boolean rings are absolutely flat: this means that every module over them is flat

Flat module

In Homological algebra, and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original...

.

Every finitely generated ideal of a Boolean ring is principal

Principal ideal

In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.More specifically:...

(indeed,

*(x,y)=(x+y+xy)*).