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Atiyah-MacDonald, exercise 2.11

The following question came up during tea today.

Let $R$ be a commutative ring with an identity and let $M \subset R^n$ be a submodule. Assume that $M \cong R^k$ for some $k$. Question : Must $k \leq n$?

If $R$ is a domain, then this is obvious. The obvious approach to proving the general result then is to mod out by the radical of $R$. If the resulting map $M / \text{rad}(R) M \rightarrow (R / \text{rad}(R))^n$ were injective, then we'd be done. However, I can't seem to prove this injectivity (I'm not even totally convinced that it's true).

Thank you for any help!