  A domain $R$ is said to have the finite factorization property if every
nonzero non-unit element of $R$ has at least one and at most finitely many
distinct factorizations up to multiplication of irreducible factors by central
units. Let $k$ be an algebraically closed field and let $A$ be a $k$-algebra.
We show that if $A$ has an associated graded ring that is a domain with the
property that the dimension of each homogeneous component is finite then $A$ is
a finite factorization domain. As a corollary, we show that many classes of
algebras have the finite factorization property, including Weyl algebras,
enveloping algebras of finite-dimensional Lie algebras, quantum affine spaces
and shift algebras. This provides a termination criterion for factorization
procedures over these algebras. In addition, we give explicit upper bounds on
the number of distinct factorizations of an element in terms of data from the
filtration.
