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 $p_1p_2\cdots p_m$ , and $q_1q_2\cdots q_n$ , 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 half-factorial rings, e.g. $\mathbb{Z}[3\sqrt{2}]$ is a HFD but not a UFD (see this paper).