half-factorial ring

An integral domain $D$ is called a half-factorial 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  $p_1{p}_2\mathrm{\cdots }{p}_m$  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 half-factorial 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).