GCD domain

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A GCD domain in mathematics is an integral domain R with the property that any two non-zero elements have a greatest common divisor (GCD). Equivalently, any two non-zero elements of R have a least common multiple (LCM). [1]

[edit] Properties

  • A unique factorization domain is a GCD domain, but the converse is not true. For example: the ring of all polynomials with rational coefficients and an integer constant term has no unique factorization since the ascending chain of principal ideals ([X], [X/2], [X/4], [X/8]...) is non-terminating, but every pair of elements has a greatest common divisor.
  • If an integral domain satisfies the ascending chain condition on principal ideals (and in particular if it is Noetherian), then it is a unique factorization domain if and only if it is a GCD domain.
  • A Bézout domain is always a GCD domain.
  • A GCD domain is integrally closed.

[edit] References

  1. ^ Scott T. Chapman, Sarah Glaz (ed.) (2000). Non-Noetherian Commutative Ring Theory, Mathematics and Its Applications (in English). Springer, 479. ISBN 0792364929. 
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