GCD domain
Encyclopedia
In mathematics, a GCD domain is an integral domain R with the property that any two non-zero elements have a greatest common divisor
(GCD). Equivalently, any two non-zero elements of R have a least common multiple
(LCM).
A GCD domain generalizes a unique factorization domain
to the non-Noetherian
setting in the following sense: an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals
(and in particular if it is Noetherian
).
, and every nonzero element is primal. In other words, every GCD domain is a Schreier domain
.
For every pair of elements x, y of a GCD domain R, a GCD d of x and y and a LCM m of x and y can be chosen such that , or stated differently, if x and y are nonzero elements and d is any GCD d of x and y, then xy/d is a LCM of x and y, and vice versa. It follows that the operations of GCD and LCM make the quotient R/~ into a distributive lattice
, where "~" denotes the equivalence relation of being associate elements.
If R is a GCD domain, then the polynomial ring R[X1,...,Xn] is also a GCD domain, and more generally, the group ring
R[G] is a GCD domain for any torsion-free commutative group G.
For a polynomial in X over a GCD domain, one can define its contents as the GCD of all its coefficients. Then the contents of a product of polynomials is the product of their contents, as expressed by Gauss's lemma
, which is valid over GCD domains.
Greatest 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...
(GCD). Equivalently, any two non-zero elements of R have a least common multiple
Least common multiple
In arithmetic and number theory, the least common multiple of two integers a and b, usually denoted by LCM, is the smallest positive integer that is a multiple of both a and b...
(LCM).
A GCD domain generalizes 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...
to the non-Noetherian
Noetherian
In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects; in particular,* Noetherian group, a group that satisfies the ascending chain condition on subgroups...
setting in the following sense: an integral domain is a UFD if and only if it is a GCD domain satisfying the ascending chain condition on principal ideals
Ascending chain condition on principal ideals
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...
(and in particular if it is Noetherian
Noetherian
In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects; in particular,* Noetherian group, a group that satisfies the ascending chain condition on subgroups...
).
Properties
Every irreducible element of a GCD domain is prime (however irreducible elements need not exist, even if the GCD domain is not a field). A GCD domain is integrally closedIntegrally closed
In mathematics, more specifically in abstract algebra, the concept of integrally closed has two meanings, one for groups and one for rings. -Commutative rings:...
, and every nonzero element is primal. In other words, every GCD domain is a Schreier domain
Schreier domain
In algebra, a Schreier domain is an integrally closed integral domain where every nonzero element is primal; i.e., whenever x divides yz, x can be written as x = x1 x2 so that x1 divides y and x2 divides z. An integral domain is said to be pre-Schreier if every nonzero element is primal. A GCD...
.
For every pair of elements x, y of a GCD domain R, a GCD d of x and y and a LCM m of x and y can be chosen such that , or stated differently, if x and y are nonzero elements and d is any GCD d of x and y, then xy/d is a LCM of x and y, and vice versa. It follows that the operations of GCD and LCM make the quotient R/~ into a distributive lattice
Distributive lattice
In mathematics, distributive lattices are lattices for which the operations of join and meet distribute over each other. The prototypical examples of such structures are collections of sets for which the lattice operations can be given by set union and intersection...
, where "~" denotes the equivalence relation of being associate elements.
If R is a GCD domain, then the polynomial ring R[X1,...,Xn] is also a GCD domain, and more generally, the group ring
Group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
R[G] is a GCD domain for any torsion-free commutative group G.
For a polynomial in X over a GCD domain, one can define its contents as the GCD of all its coefficients. Then the contents of a product of polynomials is the product of their contents, as expressed by Gauss's lemma
Gauss's lemma (polynomial)
In algebra, in the theory of polynomials , Gauss's lemma is either of two related statements about polynomials with integer coefficients:...
, which is valid over GCD domains.
Examples
- 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...
is a GCD domain. Among the GCD domains, the unique factorization domains are precisely those that are also 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...
s (which means that at least one factorization into irreducible elements exists for any nonzero nonunit). - A Bézout domainBézout domainIn mathematics, a Bézout domain 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...
(i.e., an integral domain where every finitely generated ideal is principal) is a GCD domain. Unlike principal ideal domainPrincipal ideal domainIn 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...
s (where every ideal is principal), a Bézout domain need not be a unique factorization domain; for instance 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 is a non-atomic Bézout domain, and there are many other examples. An integral domain is a PrüferPrüfer domainIn 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...
GCD domain if and only if it is a Bézout domain. - If R is a non-atomic GCD domain, then R[X] is an example of a GCD domain that is neither a unique factorization domain (since it is non-atomic) nor a Bézout domain (since X and a non-invertible and non-zero element a of R generate an ideal not containing 1, but 1 is nevertheless a GCD of X and a); more generally any ring R[X1,...,Xn] has these properties.