Throughout this entry, let be a commutative ring with .
if with and , then .
For example, is a gcd of and in any . In fact, if is a gcd of and , then . But , so that , which means that, for some , . As a result, is the unique gcd of and .
In general, however, a gcd of two elements is not unique. For example, in the ring of integers, and are both gcd’s of two relatively prime elements. By definition, any two gcd’s of a pair of elements in are associates of each other. Since the binary relation “being associates” of one anther is an equivalence relation (not a congruence relation!), we may define the gcd of and as the set
For example, as we have seen, . Also, for any , , the group of units of .
If there is no confusion, let us denote to be any element of .
If contains a unit, then and are said to be relatively prime. If is irreducible, then for any , are either relatively prime, or .
An integral domain is called a gcd domain if any two elements of , not both zero, have a gcd. In other words, is a gcd domain if for any , .
A Bezout domain is always a gcd domain. A gcd domain is a Bezout domain if for any and some .
In a gcd domain, an irreducible element is a prime element.
Given an integral domain, one can similarly define an lcm of two elements : it is an element such that and , and if is an element such that and , then . Then, a lcm domain is an integral domain such that every pair of elements has a lcm. As it turns out, the two notions are equivalent: an integral domain is lcm iff it is gcd.
The following diagram indicates how the different domains are related:
|Euclidean domain (http://planetmath.org/EuclideanRing)||PID||UFD|
|Bezout domain||gcd domain|
- 1 D. D. Anderson, Advances in Commutative Ring Theory: Extensions of Unique Factorization, A Survey, 3rd Edition, CRC Press (1999)
- 2 D. D. Anderson, Non-Noetherian Commutative Ring Theory: GCD Domains, Gauss’ Lemma, and Contents of Polynomials, Springer (2009)
- 3 D. D. Anderson (editor), Factorizations in Integral Domains, CRC Press (1997)
|Date of creation||2013-03-22 14:19:51|
|Last modified on||2013-03-22 14:19:51|
|Last modified by||CWoo (3771)|
|Defines||greatest common divisor|