TITLE: Ranks of free submodules of free modules
QUESTION [11 upvotes]: Possible Duplicate:
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!

REPLY [7 votes]: For a proof using multilinear algebra, see Corollary 5.11 at 
http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/extmod.pdf