Every irreducible is prime implies any two irreducible factorizations are equal upto ordering and associates

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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 (i.e., integral domain in which every irreducible is prime) must also satisfy the second integral domain property (i.e., integral domain in which any two irreducible factorizations are equal upto ordering and associates)
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Statement

In an integral domain with the property that every irreducible element is a prime element, it is true that any two factorizations of an element into irreducibles are equal upto ordering and associates.

Facts used

  1. Prime factorization of element is unique irreducible factorization upto ordering and associates

Proof

Fact (1) says that if a factorization into primes exists, it must equal any factorization into irreducibles, upto ordering and associates. If every irreducible is prime, then this translates to saying that any two factorizations into irreducibles are equal upto ordering and associates.