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gcd domain (Definition)

Let $ D$ be a commutative ring with $ 1\neq 0$. A gcd (greatest common divisor) of two elements $ a, b \in D$, is an element $ d \in D$ such that:

  1. $ d\mid a$ and $ d\mid b$,
  2. if $ c\in D$ with $ c\mid a$ and $ c\mid b$, then $ c\mid d$.

Any two gcd's of a pair of elements in $ D$ are associates of each other. Therefore we can speak of the gcd of $ a$ and $ b$ with the knowledge that any two such gcd's are “equivalent” by a product of a unit. The formal way of doing this is to define $ a\sim b$ iff $ a$ and $ b$ are associates, show that $ \sim$ is an equivalence relation, and form the set of equivalence classes $ D/\sim$. Alternatively, we can define

$\displaystyle \operatorname{GCD}(a,b):=\lbrace c\in D\mid c$ is a gcd of $\displaystyle a$ and $\displaystyle b\rbrace,$
and denote $ \gcd(a,b)$ to be any element of $ \operatorname{GCD}(a,b)$.

If $ \operatorname{GCD}(a,b)$ contains a unit, then $ a$ and $ b$ are said to be relatively prime. If $ a$ is irreducible, then for any $ b\in D$, $ a,b$ are either relatively prime, or $ a\mid b$.

An integral domain $ D$ is called a gcd domain if any two elements of $ D$, not both zero, have a gcd.

Remarks

The following diagram indicates how the different domains are related:
Euclidean domain $ \Longrightarrow$ PID $ \Longrightarrow$ UFD
         
    $ \Downarrow$   $ \Downarrow$
         
    Bezout domain $ \Longrightarrow$ gcd domain



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See Also: greatest common divisor, Bezout domain, divisibility in rings

Also defines:  gcd, greatest common divisor, relatively prime
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Cross-references: PID, Schreier domain, integrally closed, prime element, irreducible element, Bezout domain, converse, unique factorization domain, integral domain, irreducible, contains, equivalence classes, equivalence relation, iff, unit, product, associates, commutative ring
There are 25 references to this entry.

This is version 14 of gcd domain, born on 2004-04-23, modified 2007-05-12.
Object id is 5800, canonical name is GcdDomain.
Accessed 3559 times total.

Classification:
AMS MSC13G05 (Commutative rings and algebras :: Integral domains)

Pending Errata and Addenda
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