Let $R$ be a commutative ring with $1$ and let $S,T$ be $R$-modules such that $S$ is finitely generated and such that $S \cong S \oplus T$. Must $T=0$?
This is certainly true if $R$ is a PID, but what if $R$ is just a commutative ring with $1$? (if $R$ is a PID then the fact that $S \cong S \oplus T$ and $S$ is finitely generated implies $T$ is finitely generated as well, and then the result follows from the cancellation law for modules over a PID).