Does a finitely generated module have finitely many direct summands, up to isomorphism?

Question

One can easily see that every finitely generated abelian group has finitely many direct summands up to isomorphism.

Now, assume that $R$ is a ring and $M$ is a finitely generated $R$-module.

Does $M$ have finitely many non-trivial direct summands, up to isomorphism?

Answer 1

Let $R$ be a Dedekind domain with an infinite classgroup. If $I$ is a nonzero ideal of $R$, then $R^2\cong I\oplus I^{-1}$ as $R$-modules, where $I^{-1}$ is the fractional ideal inverse to $I$. As $R$-modules, $I\cong J$ iff $I$ and $J$ are in the same ideal classes. Therefore there are infinitely many direct summands in $R^2$ which are not isomorphic as modules.