  A ring has bounded factorizations if every cancellative nonunit $a \in R$ can
be written as a product of atoms and there is a bound $\lambda(a)$ on the
lengths of such factorizations. The bounded factorization property is one of
the most basic finiteness properties in the study of non-unique factorizations.
Every commutative noetherian domain has bounded factorizations, but it is open
whether such a result holds in the noncommutative setting. We provide
sufficient conditions for a noncommutative noetherian prime ring to have
bounded factorizations. Moreover, we construct a (noncommutative) finitely
presented semigroup algebra that is an atomic domain but does not satisfy the
ascending chain condition on principal right or left ideals (ACCP), whence it
does not have bounded factorizations.
