GCD domain
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In mathematics, an integral domain is called a GCD domain if any two non-zero elements have a greatest common divisor (GCD).
[edit] Properties
- A unique factorization domain is a GCD domain, but the converse is not true. The ring of all polynomials with rational coefficients and an integer constant 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. However, if an integral domain satisfies the ascending chain condition on principal ideals, then it has unique factorization if and only if it is a GCD domain.
- A Bézout domain is always a GCD domain.
- A GCD domain is integrally closed.
