Bezout implies gcd

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This article gives the statement and possibly, proof, of an implication relation between two integral domain properties. That is, it states that every integral domain satisfying the first integral domain property must also satisfy the second integral domain property
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Statement

Property-theoretic statement

The property of integral domains of being a Bezout domain, is stronger than the property of being a gcd domain.

Proof

Proof outline

  • Pick two elements in the given Bezout domain. We need to determine their gcd.
  • Consider the ideal generated by these two elements. This is a finitely generated ideal, and by the definition of Bezout domain, is a principal ideal. Pick a generator for this principal ideal.
  • Show that this generator is indeed the gcd of the two elements.