Bézout domain
Encyclopedia
In mathematics
, a Bézout domain is an integral domain in which the sum of two principal ideal
s is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every finitely generated ideal is principal. Any principal ideal domain
(PID) is a Bézout domain, but a Bézout domain need not be Noetherian ring
, so it could have non-finitely generated ideals (which obviously excludes being a PID); if so, it is not a unique factorization domain
(UFD), but still a GCD domain
. The theory of Bézout domains retains many of the properties of PIDs, without requiring the Noetherian property. Bézout domains are named after the French
mathematician
Étienne Bézout
.
that is a linear combination
of them: this is equivalent to the statement that an ideal which is generated by two elements is also generated by a single element, and induction demonstrates that all finitely generated ideals are principal. The expression of the greatest common divisor of two elements of a PID as a linear combination is often called Bézout's identity
, whence the terminology.
Note that the above gcd condition is stronger than the mere existence of a gcd. An integral domain where a gcd exists for any two elements is called a GCD domain
and thus Bézout domains are GCD domains. In particular, in a Bézout domain, irreducibles
are prime
(but as the algebraic integer example shows, they need not exist).
For a Bézout domain R, the following conditions are all equivalent:
The equivalence of (1) and (2) was noted above. Since a Bézout domain is a GCD domain, it follows immediately that (3), (4) and (5) are equivalent. Finally, if R is not Noetherian, then there exists an infinite ascending chain of finitely generated ideals, so in a Bézout domain an infinite ascending chain of principal ideals. (4) and (2) are thus equivalent.
A Bézout domain is a Prüfer domain
, i.e., a domain in which each finitely generated ideal is invertible, or said another way, a commutative semihereditary
domain.)
Roughly speaking, one may view the implications "Bézout domain implies Prüfer domain and GCD-domain" as the non-Noetherian analogues of the more familiar "PID implies Dedekind domain
and
UFD". The analogy fails to be precise in that a UFD (or an atomic Prüfer domain) need not be Noetherian.
Prüfer domains can be characterized as integral domains whose localization
s at all prime
(equivalently, at all maximal
) ideals are valuation domains
. So the localization of a Bézout domain at a prime ideal is a valuation domain. Since an invertible ideal in a local ring
is principal, a local ring is a Bézout domain iff it is a valuation domain. Moreover a valuation domain with noncyclic (equivalently non-discrete) value group is not Noetherian, and every totally ordered
abelian group
is the value group of some valuation domain. This gives many examples of non-Noetherian Bézout domains.
In noncommutative algebra, right Bézout domains are domains whose finitely generated right ideals are principal right ideals, that is, of the form xR for some x in R. One notable result is that a right Bézout domain is a right Ore domain. This fact is not interesting in the commutative case, since every commutative domain is an Ore domain. Right Bézout domains are also right semihereditary rings.
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 Bézout domain is an integral domain in which the sum of two principal ideal
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:...
s is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every finitely generated ideal is principal. Any principal ideal domain
Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
(PID) is a Bézout domain, but a Bézout domain need not be Noetherian ring
Noetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
, so it could have non-finitely generated ideals (which obviously excludes being a PID); if so, it is not a unique factorization domain
Unique factorization domain
In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
(UFD), but still a GCD domain
GCD domain
In mathematics, a GCD domain is an integral domain R with the property that any two non-zero elements have a greatest common divisor . Equivalently, any two non-zero elements of R have a least common multiple ....
. The theory of Bézout domains retains many of the properties of PIDs, without requiring the Noetherian property. Bézout domains are named after the French
French people
The French are a nation that share a common French culture and speak the French language as a mother tongue. Historically, the French population are descended from peoples of Celtic, Latin and Germanic origin, and are today a mixture of several ethnic groups...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Étienne Bézout
Étienne Bézout
-External links:...
.
Examples
- All PIDs are Bézout domains.
- Examples of Bézout domains that are not PIDs include the ring of entire functionEntire functionIn complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
s (functions holomorphic on the whole complex plane) and the ring of all algebraic integerAlgebraic integerIn number theory, an algebraic integer is a complex number that is a root of some monic polynomial with coefficients in . The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A...
s. In case of entire functions, the only irreducible elements are functions associated to a polynomial function of degree 1, so an element has a factorization only if it has finitely many zeroes. In the case of the algebraic integers there are no irreducible elements at all, since for any algebraic integer its square root (for instance) is also an algebraic integer. This shows in both cases that the ring is not a UFD, and so certainly not a PID. - The following general construction produces a Bézout domain S that is not a UFD from any Bézout domain R that is not a field, for instance from a PID; the case is the basic example to have in mind. Let F be the field of fractionsField of fractionsIn abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...
of R, and put , the subring F[X] of polynomials whose constant term lies in R. This ring is not Noetherian, since an element like X with zero constant term can be divided indefinitely by noninvertible elements of R, which are still noninvertible in S, and the ideal generated by all these quotients of is not finitely generated (and so X has no factorization in S). One shows as follows that S is a Bézout domain.
-
- It suffices to prove that for every pair a, b in S there exist s,t in S such that divides both a and b.
- If a and b have a common divisor d, it suffices to prove this for a/d and b/d, since the same s,t will do.
- We may assume the polynomials a and b nonzero; if both have a zero constant term, then let n be the minimal exponent such that at least one of them has a nonzero coefficient of Xn; one can find f in F such that fXn is a common divisor of a and b and divide by it.
- We may therefore assume at least one of a, b has a nonzero constant term. If a and b viewed as elements of F[X] are not relatively prime, there is a greatest common divisor of a and b in this UFD that has constant term 1, and therefore lies in S; we can divide by this factor.
- We may therefore also assume that a and b are relatively prime in F[X], so that 1 lies in , and some constant polynomial r in R lies in . Also, since R is a Bézout domain, the gcd d in R of the constant terms a0 and b0 lies in , and applying its Bézout coefficients to a and b gives a polynomial p in with constant term d. Therefore has a zero constant term, and as a consequence is a multiple in S of the constant polynomial r, and in particular lies in . But then d does as well, and it is a common divisor of a and b in S, which completes the proof.
Properties
A ring is a Bézout domain if and only if it is an integral domain in which any two elements have a greatest common divisorGreatest common divisor
In mathematics, the greatest common divisor , also known as the greatest common factor , or highest common factor , of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.For example, the GCD of 8 and 12 is 4.This notion can be extended to...
that is a linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
of them: this is equivalent to the statement that an ideal which is generated by two elements is also generated by a single element, and induction demonstrates that all finitely generated ideals are principal. The expression of the greatest common divisor of two elements of a PID as a linear combination is often called Bézout's identity
Bézout's identity
In number theory, Bézout's identity for two integers a, b is an expressionwhere x and y are integers , such that d is a common divisor of a and b. Bézout's lemma states that such coefficients exist for every pair of nonzero integers...
, whence the terminology.
Note that the above gcd condition is stronger than the mere existence of a gcd. An integral domain where a gcd exists for any two elements is called a GCD domain
GCD domain
In mathematics, a GCD domain is an integral domain R with the property that any two non-zero elements have a greatest common divisor . Equivalently, any two non-zero elements of R have a least common multiple ....
and thus Bézout domains are GCD domains. In particular, in a Bézout domain, irreducibles
Irreducible element
In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.Irreducible elements should not be confused with prime elements...
are prime
Prime element
In abstract algebra, an element p of a commutative ring R is said to be prime if it is not zero, not a unit and whenever p divides ab for some a and b in R, then p divides a or p divides b...
(but as the algebraic integer example shows, they need not exist).
For a Bézout domain R, the following conditions are all equivalent:
- R is a principal ideal domain.
- R is Noetherian.
- R is a unique factorization domainUnique factorization domainIn mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
(UFD). - R satisfies the ascending chain condition on principal ideals (ACCP).
- Every nonzero nonunit in R factors into a product of irreducibles (R is an atomic domainAtomic domainIn mathematics, more specifically ring theory, an atomic domain or factorization domain is an integral domain, every non-zero non-unit of which can be written as a product of irreducible elements...
).
The equivalence of (1) and (2) was noted above. Since a Bézout domain is a GCD domain, it follows immediately that (3), (4) and (5) are equivalent. Finally, if R is not Noetherian, then there exists an infinite ascending chain of finitely generated ideals, so in a Bézout domain an infinite ascending chain of principal ideals. (4) and (2) are thus equivalent.
A Bézout domain is a Prüfer domain
Prüfer domain
In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules...
, i.e., a domain in which each finitely generated ideal is invertible, or said another way, a commutative semihereditary
Hereditary ring
In mathematics, especially in the area of abstract algebra known as module theory, a ring R is called hereditary if all submodules of projective modules over R are again projective...
domain.)
Roughly speaking, one may view the implications "Bézout domain implies Prüfer domain and GCD-domain" as the non-Noetherian analogues of the more familiar "PID implies Dedekind domain
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...
and
UFD". The analogy fails to be precise in that a UFD (or an atomic Prüfer domain) need not be Noetherian.
Prüfer domains can be characterized as integral domains whose localization
Localization of a ring
In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
s at all prime
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...
(equivalently, at all maximal
Maximal 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...
) ideals are valuation domains
Valuation ring
In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D....
. So the localization of a Bézout domain at a prime ideal is a valuation domain. Since an invertible ideal in a local ring
Local ring
In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
is principal, a local ring is a Bézout domain iff it is a valuation domain. Moreover a valuation domain with noncyclic (equivalently non-discrete) value group is not Noetherian, and every totally ordered
Total order
In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
is the value group of some valuation domain. This gives many examples of non-Noetherian Bézout domains.
In noncommutative algebra, right Bézout domains are domains whose finitely generated right ideals are principal right ideals, that is, of the form xR for some x in R. One notable result is that a right Bézout domain is a right Ore domain. This fact is not interesting in the commutative case, since every commutative domain is an Ore domain. Right Bézout domains are also right semihereditary rings.