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Bezout domain (Definition)

A Bezout domain $ D$ is an integral domain such that every finitely generated ideal of $ D$ is principal.

Remarks.

  • A PID is obviously a Bezout domain.
  • Furthermore, a Bezout domain is a gcd domain. To see this, suppose $ D$ is a Bezout domain with $ a,b\in D$. By definition, there is a $ d\in D$ such that $ (d)=(a,b)$, the ideal generated by $ a$ and $ b$. So $ a\in (d)$ and $ b\in (d)$ and therefore, $ d\mid a$ and $ d\mid b$. Next, suppose $ c\in D$ and that $ c\mid a$ and $ c\mid b$. Then both $ a,b\in (c)$ and so $ d\in (c)$. This means that $ c\mid d$ and we are done.
  • From the discussion above, we see in a Bezout domain $ D$, a greatest common divisor exists for every pair of elements. Furthermore, if $ \operatorname{gcd}(a,b)$ denotes one such greatest common divisor between $ a,b\in D$, then for some $ r,s\in D$:
    $\displaystyle \operatorname{gcd}(a,b)=ra+sb.$
    The above equation is known as the Bezout identity, or Bezout's Lemma.



"Bezout domain" is owned by CWoo.
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See Also: gcd domain, divisibility by product

Other names:  Bézout domain
Also defines:  Bezout identity

Attachments:
example of a Bezout domain that is not a PID (Example) by Wkbj79
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Cross-references: Bezout's lemma, equation, greatest common divisor, ideal generated by, gcd domain, PID, ideal, finitely generated, integral domain
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This is version 7 of Bezout domain, born on 2004-04-23, modified 2005-01-28.
Object id is 5801, canonical name is BezoutDomain.
Accessed 4697 times total.

Classification:
AMS MSC: 13G05 (Commutative rings and algebras :: Integral domains)
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