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Fundamental theorem of arithmetics: A principal ideal domain is factorial. i.e: Any non zero element of a principal ideal domain can be decomposed in a unique product of irreductibles.

The structure of the proof goes like this::

  • Proof of the existence of a decomposition: In this part, it is shown any non-zero and non-invertible elements of a principal ideal domain can be decomposed in a product of irrreductibles.

  • Proof of the uniqueness

My problem: In the first part they show that any non-zero and non-invertible elements of a principal ideal domain can be decomposed in a product of irrreductibles. What about non-zero but invertible elements of a principal ideal domain?

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In a commutative ring $R$, say that $a\sim b$ if there exists a unit $u$ such that $a=bu$. ($a$ is said to be 'an associate to $b$'.) This turns out to be an equivalence relation.

In the theory of unique factorization, when you say a particular factorziaton is a "unique product," it is subject to the caveat that things are unique up to being associates.

Now you can verify that the set of units of the ring forms an equivalence class in this relation, so that every unit is equivalent to $1$. Therefore, it's pointless to talk about factorizations of units because all units are "the same as $1$."

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