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halffactorial ring
An integral domain $D$ is called a halffactorial ring (HFD) if it satisfies the following conditions:

Every nonzero element of $D$ that is not a unit can be factored into a product of a finite number of irreducibles.

If $p_{1}p_{2}\cdots p_{m}$ and $q_{1}q_{2}\cdots q_{n}$ are two factorizations of the same element $a$ into irreducibles, then $m=n$.
If, in addition, the irreducibles $p_{i}$ and $q_{j}$ are always pairwise associates, then $D$ is a factorial ring (UFD).
For example, many orders in the maximal order of an algebraic number field are halffactorial rings, e.g. $\mathbb{Z}[3\sqrt{2}]$ is a HFD but not a UFD (see this paper).
Defines:
HFD
Synonym:
halffactorial domain
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Definition
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