Exercise A0.III.6.02

Proposition

For some ring $R$, there exists a non-trivial $R$-module $M$ such that $M \cong M \oplus M.$

Proof

Consider the example of an abelian group isomorphic to its own direct square constructed in II.3.4. Since abelian groups are simply $\mathbb{Z}$-modules, this example needs no further justification.