Let f:R→S be a homomorphism of commutative rings. Many techniques for studying R-modules focus on finitely generated modules. As a consequence, these techniques are not well-suited for studying S as an R-module. However, a technique of Avramov, Foxby, and Herzog sometimes allows one to replace the original homomorphism with a surjective one R′→S where R and R′ are tightly connected. In this setting, S is a cyclic R′-module, so one can study it using finitely generated techniques. I will give a general introduction to such factorizations, followed by a discussion of some new results on ``weakly functorial properties’’ of such factorizations and applications. The new results are joint with Sean Sather-Wagstaff.