DG module structures and minimal free resolutions modulo an exact zero-divisor

Let $Q$ be a local ring with maximal ideal $\mathfrak{n}$ and let $f,g\in \mathfrak{n}\setminus {\mathfrak{n}}^2$ with $fg=0$. When $M$ is a finite $Q$-module with $fM=0$, we show that a minimal free resolution of $M$ over $Q$ has a differential graded module structure over the differential graded algebra $Q⟨y,t\mid \mathrm{\partial }(y)=f,\mathrm{\partial }(t)=gy⟩$. When $(f,g)$ is a pair of exact zero divisors, we use this structure to describe a minimal free resolution of $M$ over $Q\mathrm{/}(f)$.