UFD 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 unique factorization domain is stronger than the property of being a gcd domain.

Verbal statement

Any unique factorization domain is a gcd domain.

Definitions used

Unique factorization domain

A unique factorization domain is an integral domain where every element can be factorized uniquely as a product of irreducible elements.

gcd domain

A gcd domain is an integral domain for which, given any two elements, there is an element, called the gcd of the two elements, such that any common divisor of the two elements divides the gcd.

Proof

Proof outline

  • Pick two elements in the UFD. We need to exhibit an element which satisfies the property demanded of the gcd of these two elements
  • Write down unique factorizations of both elements
  • For each prime, pick as its gcd exponent the minimum of its exponents in the unique factorizations of both elements. The gcd is then the element where the exponent of each prime is its gcd exponent.