Let $F$ be a field. Why is $\frac{F[x]}{(x-1)}\oplus \frac{F[x]}{(x-1)}$ not isomorphic to $\frac{F[x]}{(x-1)^2}$ as $F[x]$-modules?
Let $R$ be a PID, and let $\pi\in R$ be prime. Is it $R/\langle\pi\rangle \times R/\langle \pi\rangle$ a cyclic $R$-module?