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For the sentence
An integral domain $R$ is a unique factorization domain (UFD for short) if every non-unit has a factorization $u = r_1 \cdots r_n$ as product of irreducible non-units and this decomposition is unique up to renumbering and rescaling the irreducibles by units.
should the “product of irreducible non-units” be “arbitrary/infinitary product” or “finite product”?
Precised that $0$ is never irreducible as a consequence of the definition of irreducible.
Corrected the definition:
The unique factorization condition if for non-zero non-units. With the current definition, there doesn’t exist any UFD. If zero must have a decomposition as a product of irreducibles, then we have $0 = r_{1}...r_{n}$ and thus one of the $r_{i}$ is equal to $0$ because we are in an integral domain. But $0$ is never irreducible, absurd.
Replaced “irreducible non unit” by “irreducible” because it is redundant.
Precised that $n$ must be greater than $1$ in the decomposition.
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