Bézout domain

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In mathematics, a Bézout domain is a form of a Prüfer domain. It is an integral domain in which the sum of two principal ideals 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 a 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 is 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.

Mathematics field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

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. Prüfer domains are named after the German mathematician Heinz Prüfer.

In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c.

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Examples

In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic at all finite points over the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function f(z) has a root at w, then f(z)/(z−w), taking the limit value at w, is an entire function. On the other hand, neither the natural logarithm nor the square root is an entire function, nor can they be continued analytically to an entire function.

In algebraic number theory, an algebraic integer is a complex number that is a root of some monic polynomial with coefficients in ℤ. The set of all algebraic integers, A, is closed under addition and multiplication and therefore is a commutative subring of the complex numbers. The ring A is the integral closure of regular integers ℤ in complex numbers.

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.

  1. It suffices to prove that for every pair a, b in S there exist s, t in S such that as + bt divides both a and b.
  2. 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.
  3. 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.
  4. 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.
  5. We may therefore also assume that a and b are relatively prime in F[X], so that 1 lies in aF[X] + bF[X], and some constant polynomial r in R lies in aS + bS. Also, since R is a Bézout domain, the gcd d in R of the constant terms a0 and b0 lies in a0R + b0R. Since any element without constant term, like a − a0 or b − b0, is divisible by any nonzero constant, the constant d is a common divisor in S of a and b; we shall show it is in fact a greatest common divisor by showing that it lies in aS + bS. Multiplying a and b respectively by the Bézout coefficients for d with respect to a0 and b0 gives a polynomial p in aS + bS with constant term d. Then p − d has a zero constant term, and so is a multiple in S of the constant polynomial r, and therefore lies in aS + bS. But then d does as well, 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 divisor 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.

In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For example, the gcd of 8 and 12 is 4.

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. The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.

In elementary number theory, Bézout's identity is the following theorem:

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).

In mathematics, a GCD domain is an integral domain R with the property that any two elements have a greatest common divisor (GCD); i.e., there is a unique minimal principal ideal containing the ideal generated by given two elements. Equivalently, any two elements of R have a least common multiple (LCM).

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.

In mathematics, specifically in abstract algebra, a prime element of a commutative ring is an object satisfying certain properties similar to the prime numbers in the integers and to irreducible polynomials. Care should be taken to distinguish prime elements from irreducible elements, a concept which is the same in UFDs but not the same in general.

For a Bézout domain R, the following conditions are all equivalent:

  1. R is a principal ideal domain.
  2. R is Noetherian.
  3. R is a unique factorization domain (UFD).
  4. R satisfies the ascending chain condition on principal ideals (ACCP).
  5. Every nonzero nonunit in R factors into a product of irreducibles (R is an atomic domain).

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.)

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. If this is required only for finitely generated submodules, it is called semihereditary.

Consequently, one may view the equivalence "Bézout domain iff Prüfer domain and GCD-domain" as analogous to the more familiar "PID iff Dedekind domain and UFD".

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. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below.

Prüfer domains can be characterized as integral domains whose localizations 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.

Modules over a Bézout domain

Some facts about modules over a PID extend to modules over a Bézout domain. Let R be a Bézout domain and M finitely generated module over R. Then M is flat if and only if it is torsion-free. [2]

See also

Related Research Articles

In mathematics, more specifically in ring theory, a Euclidean domain is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of the integers. This generalized Euclidean algorithm can be put to many of the same uses as Euclid's original algorithm in the ring of integers: in any Euclidean domain, one can apply the Euclidean algorithm to compute the greatest common divisor of any two elements. In particular, the greatest common divisor of any two elements exists and can be written as a linear combination of them. Also every ideal in a Euclidean domain is principal, which implies a suitable generalization of the fundamental theorem of arithmetic: every Euclidean domain is a unique factorization domain.

Prime ideal object in mathematics

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal.

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 principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot.

Ring (mathematics) algebraic structure in mathematics, not necessarily with multiplicative identity

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series, matrices and functions.

In mathematics, more specifically in the area of abstract algebra known as ring theory, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals, which means there is no infinite ascending sequence of left ideals; that is, given any chain of left ideals,

In mathematics, a unique factorization domain (UFD) is an integral domain in which every non-zero non-unit element can be written as a product of prime elements, uniquely up to order and units, analogous to the fundamental theorem of arithmetic for the integers. UFDs are sometimes called factorial rings, following the terminology of Bourbaki.

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative rings where multiplication is not required to be commutative.

In the mathematical field of ring theory, a principal ideal is an ideal in a ring that is generated by a single element of through multiplication by every element of . The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset generated by a single element , which is to say the set of all elements less than or equal to in .

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a statement about polynomials over the integers, or, more generally, over a unique factorization domain. Gauss's lemma underlies all the theory of factorization and greatest common divisors of such polynomials.

In mathematics, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. When this is satisfied for both left and right ideals, such as the case when R is a commutative ring, R can be called a principal ideal ring, or simply principal ring.

In commutative algebra, a Krull ring or Krull domain is a commutative ring with a well behaved theory of prime factorization. They were introduced by Wolfgang Krull (1931). They are a higher-dimensional generalization of Dedekind domains, which are exactly the Krull domains of dimension at most 1.

In commutative algebra, an integrally closed domainA is an integral domain whose integral closure in its field of fractions is A itself. Many well-studied domains are integrally closed: Fields, the ring of integers Z, unique factorization domains and regular local rings are all integrally closed.

In abstract algebra, the ascending chain condition can be applied to the posets of principal left, principal right, or principal two-sided ideals of a ring, partially ordered by inclusion. The ascending chain condition on principal ideals is satisfied if there is no infinite strictly ascending chain of principal ideals of the given type (left/right/two-sided) in the ring, or said another way, every ascending chain is eventually constant.

This is a glossary of commutative algebra.

References

  1. ↑ Cohn
  2. ↑ Bourbaki 1989 , Ch I, §2, no 4, Proposition 3
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