An unrestricted notion of the finite factorization property

A nonzero element of an integral domain (or commutative cancellative monoid) is called atomic if it can be written as a finite product of irreducible elements (also called atoms). In this paper, we introduce and investigate an unrestricted version of the finite factorization property, extending the work on unrestricted UFDs carried out by Coykendall and Zafrullah who first studied unrestricted. An integral domain is said to have the unrestricted finite factorization (U-FF) property if every atomic element has only finitely many factorizations, or equivalently, if its atomic subring is a finite factorization domain (FFD). We position the property U-FF within the hierarchy of classical finiteness conditions, showing that every IDF domain is U-FF but not conversely, and we analyze its behavior under standard constructions. In particular, we determine necessary and sufficient conditions for the U-FF property to ascend along $D+M$ extensions, prove that nearly atomic IDF domains are FFDs, and construct an explicit example of an integral domain with the U-FF property whose polynomial ring is not U-FF. These results demonstrate that the U-FF property behaves analogously to the IDF property, while providing a finer interpolation between the IDF and the FF conditions.