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

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

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If ${p}_{1}{p}_{2}\mathrm{\cdots}{p}_{m}$ and ${q}_{1}{q}_{2}\mathrm{\cdots}{q}_{n}$ are two factorizations of the same element $a$ into irreducibles, then $m=n$.
If, in , the irreducibles ${p}_{i}$ and ${q}_{j}$ are always pairwise associates^{}, then $D$ is a factorial ring (UFD).
For example, many orders (http://planetmath.org/OrderInAnAlgebra) 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 http://www.math.ndsu.nodak.edu/faculty/coykenda/paper6b.pdfthis paper).
Title  halffactorial ring 

Canonical name  HalffactorialRing 
Date of creation  20130322 18:31:14 
Last modified on  20130322 18:31:14 
Owner  pahio (2872) 
Last modified by  pahio (2872) 
Numerical id  7 
Author  pahio (2872) 
Entry type  Definition 
Classification  msc 13G05 
Synonym  halffactorial domain 
Defines  HFD 