Flatness criterion over dual numbers ($M\text{ flat over }k[\epsilon ]/({\epsilon }^2)⟺\ker \epsilon =\epsilon M$)

For $D=k[\epsilon ]/({\epsilon }^2)$, a module $M$ is a flat module exactly when $\ker (\epsilon :M\to M)=\epsilon M$. Flatness implies this by tensoring the exact sequence $0\to (\epsilon )\to D\to k\to 0$. Conversely lift a $k$-basis of $M/\epsilon M$ to $M$. Every element is a finite combination of the lifts plus $\epsilon$ times another such combination, proving generation over $D$. A relation first reduces to zero coefficients modulo $\epsilon$; the kernel condition then reduces its remaining coefficients to zero as well. The lifts form a free $D$-basis, proving flatness.