In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where ...
containing the ideal generated by two given elements. Equivalently, any two elements of ''R'' have a
least common multiple
In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers ''a'' and ''b'', usually denoted by lcm(''a'', ''b''), is the smallest positive integer that is divisible by ...
(LCM).
A GCD domain generalizes a
unique factorization domain (UFD) to a non-
Noetherian setting in the following sense: an integral domain is a UFD
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
it is a GCD domain satisfying the
ascending chain condition on principal ideals (and in particular if it is
Noetherian).
GCD domains appear in the following chain of
class inclusions:
Properties
Every
irreducible element of a GCD domain is
prime
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
. A GCD domain is
integrally closed, 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 an 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 an LCM of ''x'' and ''y'', and vice versa. It
follows
Follows is a surname. Notable people with the surname include:
* Dave Follows (1941–2003), British cartoonist
* Denis Follows (1908–1983), British sports administrator
* Geoffrey Follows (1896–1983), British colonial administrator
* Megan Fo ...
that the operations of GCD and LCM make the quotient ''R''/~ into a
distributive lattice, where "~" denotes the equivalence relation of being
associate elements
In mathematics, specifically 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 se ...
. The equivalence between the existence of GCDs and the existence of LCMs is not a corollary of the similar result on
complete lattice
In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' ...
s, as the quotient ''R''/~ need not be a complete lattice for a GCD domain ''R''.
If ''R'' is a GCD domain, then the polynomial ring ''R''
1,...,''X''''n''">'X''1,...,''X''''n''is also a GCD domain.
R is a GCD domain if and only if finite intersections of its
principal ideal
In mathematics, specifically ring theory, a principal ideal is an ideal I in a ring R that is generated by a single element a of R through multiplication by every element of R. The term also has another, similar meaning in order theory, where ...
s are principal. In particular,
, where
is the LCM of
and
.
For a polynomial in ''X'' over a GCD domain, one can define its content as the GCD of all its coefficients. Then the content of a product of polynomials is the product of their contents, as expressed by
Gauss's lemma, which is valid over GCD domains.
Examples
*A
unique factorization domain is a GCD domain. Among the GCD domains, the unique factorization domains are precisely those that are also
atomic domains (which means that at least one factorization into irreducible elements exists for any nonzero nonunit).
*A
Bézout domain (i.e., an integral domain where every finitely generated ideal is principal) is a GCD domain. Unlike
principal ideal domains (where ''every'' ideal is principal), a Bézout domain need not be a unique factorization domain; for instance the ring of
entire functions is a non-atomic Bézout domain, and there are many other examples. An integral domain is a
Prüfer Pruefer or Prüfer is a surname of German derivation, and may refer to:
* Heinz Prüfer
Ernst Paul Heinz Prüfer (10 November 1896 – 7 April 1934) was a German Jewish mathematician born in Wilhelmshaven. His major contributions were on abelian ...
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''
1,...,''X''''n''">'X''1,...,''X''''n''has these properties.
*A
commutative monoid ring