Let $A$ be the ring

\[ A = \{ f\in k[x] \mid f(0) = f(1) \} . \]Show there exists a nontrivial invertible $A$-module, unless $k = {\mathbf F}_2$. (Hint: Think about $\mathop{\mathrm{Spec}}(A)$ as identifying $0$ and $1$ in ${\mathbf A}^1_ k = \mathop{\mathrm{Spec}}(k[x])$.)

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