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
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 set ...
''R'' with the property that any two elements have a
greatest common divisor
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 two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...
(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 it ...
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 bot ...
(LCM).
A GCD domain generalizes a
unique factorization domain
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an ...
(UFD) to a non-
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
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 bicondi ...
it is a GCD domain satisfying the
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 (mathematics), ring, partially ordered by inclusion (set theory), inclusion. The ascending ch ...
(and in particular if it is
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
).
GCD domains appear in the following chain of
class inclusions:
Properties
Every
irreducible element
In algebra, an irreducible element of a domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements.
Relationship with prime elements
Irreducible elements should not be confus ...
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
Primal may refer to:
Psychotherapy
* ''Primal'', the core concept in primal therapy, denotes the full reliving and cathartic release of an early traumatic experience
* Primal scene (in psychoanalysis), refers to the witnessing by a young child of ...
. 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 ...
that the operations of GCD and LCM make the quotient ''R''/~ into a
distributive lattice
In mathematics, a distributive lattice is a lattice in 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 uni ...
, 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 sett ...
. 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 it ...
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
In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an ...
is a GCD domain. Among the GCD domains, the unique factorization domains are precisely those that are also
atomic domain In mathematics, more specifically ring theory, an atomic domain or factorization domain is an integral domain in which every non-zero non-unit can be written in at least one way as a finite product of irreducible elements. Atomic domains are diff ...
s (which means that at least one factorization into irreducible elements exists for any nonzero nonunit).
*A
Bézout domain 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 fini ...
(i.e., an integral domain where every finitely generated ideal is principal) is a GCD domain. Unlike
principal ideal domain
In mathematics, 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, ...
s (where ''every'' ideal is principal), a Bézout domain need not be a unique factorization domain; for instance the ring of
entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any fin ...
s 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, German Jewish mathematician (1896-1934)
* Kevin Prufer, American poet (1969)
* Gustav Franz Pruefer, American Music instrument inventor (1861-1951)
See a ...
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
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
monoid ring
In abstract algebra, a monoid ring is a ring constructed from a ring and a monoid, just as a group ring is constructed from a ring and a group.
Definition
Let ''R'' be a ring and let ''G'' be a monoid. The monoid ring or monoid algebra of ''G'' o ...