In Frayleigh the definition of an UFD (unique factorization domain) is the following:
An integral domain $D$ is a unique factorization domain if the following conditions are satisfied:
- Every element of $D$ that is neither $0$ nor a unit can be factored into a product of a finite number of irreducibles.
- If $p_1\ldots p_r$ and $q_1\ldots q_s$ are two factorizations of the same element of $D$ into irreducibles, then $r = s$ and $q_j$ can be renumbered so that $p_i$ and $q_i$ are associates.
Now suppose that $ab = c^3$ in an $D$ which is an UFD and suppose that the only common factors of $a,b$ are units. Considering the facotorization of $a,b,c$, I deduce that each irreducible factor in $a$ must have multiplicity $3$ (since non of its factors is associated to the ones in $b$), however, this leaves me with $$a = up_1^3\ldots p_k^3$$ with $u$ some unit in $D$. Can we say something about this unit? (More specifically: can I say that this unit must be a third power of some other unit?