Template:Short description In mathematics, a Boolean ring Template:Math is a ring for which Template:Math for all Template:Math in Template:Math, that is, a ring that consists of only idempotent elements.Template:SfnTemplate:SfnTemplate:Sfn An example is the ring of integers modulo 2.
Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet Template:Math, and ring addition to exclusive disjunction or symmetric difference (not disjunction Template:Math,Template:Refn which would constitute a semiring). Conversely, every Boolean algebra gives rise to a Boolean ring. Boolean rings are named after the founder of Boolean algebra, George Boole.
NotationEdit
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 Template:Math for the ring sum of Template:Math and Template:Math, and use Template:Math for their product.
- In logic, a common notation is to use Template:Math for the meet (same as the ring product) and use Template:Math for the join, given in terms of ring notation (given just above) by Template:Math.
- In set theory and logic it is also common to use Template:Math for the meet, and Template:Math for the join Template:Math.Template:Sfn This use of Template:Math is different from the use in ring theory.
- A rare convention is to use Template:Math for the product and Template:Math for the ring sum, in an effort to avoid the ambiguity of Template:Math.
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 field of two elements: otherwise there cannot be a (unital) ring homomorphism of the field of two elements into the Boolean ring. (This is the same as the old use of the terms "ring" and "algebra" in measure theory.Template:Efn)
ExamplesEdit
One example of a Boolean ring is the power set of any set Template:Math, 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 Template:Math, 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 algebrasEdit
Since the join operation Template:Math in a Boolean algebra is often written additively, it makes sense in this context to denote ring addition by Template:Math, a symbol that is often used to denote exclusive or.
Given a Boolean ring Template:Math, for Template:Math and Template:Math in Template:Math we can define
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.
Properties of Boolean ringsEdit
Every Boolean ring Template:Math satisfies Template:Math for all Template:Math in Template:Math, because we know
and since Template:Math is an abelian group, we can subtract Template:Math from both sides of this equation, which gives Template:Math. A similar proof shows that every Boolean ring is commutative:
- Template:Math and this yields Template:Math, which means Template:Math (using the first property above).
The property Template:Math shows that any Boolean ring is an associative algebra over the field Template:Math with two elements, in precisely one way.Template:Citation needed In particular, any finite Boolean ring has as cardinality a power of two. Not every unital associative algebra over Template:Math is a Boolean ring: consider for instance the polynomial ring Template:Math.
The quotient ring Template:Math of any Boolean ring Template:Math modulo any ideal Template:Math is again a Boolean ring. Likewise, any subring of a Boolean ring is a Boolean ring.
Any localization Template:Math of a Boolean ring Template:Math by a set Template:Math is a Boolean ring, since every element in the localization is idempotent.
The maximal ring of quotients Template:Math (in the sense of Utumi and Lambek) of a Boolean ring Template:Math is a Boolean ring, since every partial endomorphism is idempotent.Template:Sfn
Every prime ideal Template:Math in a Boolean ring Template:Math is maximal: the quotient ring Template:Math is an integral domain and also a Boolean ring, so it is isomorphic to the field Template:Math, which shows the maximality of Template:Math. 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, Template:Math. 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.
UnificationEdit
Unification in Boolean rings is decidable,Template:Sfn that is, algorithms exist to solve arbitrary equations over Boolean rings. Both unification and matching in finitely generated free Boolean rings are NP-complete, and both are NP-hard in finitely presented Boolean rings.Template:Sfn (In fact, as any unification problem Template:Math in a Boolean ring can be rewritten as the matching problem Template:Math, the problems are equivalent.)
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).Template:Sfn
See alsoEdit
NotesEdit
CitationsEdit
ReferencesEdit
- Template:Citation
- Template:Cite journal
- Template:Cite journal
- Template:Citation
- Template:Citation
- Template:Cite book
- Template:Cite book
- Template:Cite book
- Template:Citation
- Template:Eom
External linksEdit
- John Armstrong, Boolean Rings