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Let $R$ be a nonzero commutative ring, and let $n,m$ be integers. Let $f:R^n\rightarrow R^m$ be an injective homomorphism of $R$-modules.

I'm trying to show that $n\leq m$.

My idea: Assume that $n>m$, let $i:R^m\rightarrow R^n$ be the "embedding in the first $n$ coordinates", and take the composition $g:R^n\rightarrow R^n$ define by $g=i\circ f$.

First, $g$ is injective. Second, we can apply a version of Nakayama's lemma to get a monic polynomial $p(X)\in R[X]$ such that $p(g)=0$.

Now, I somehow want to use the fact that the image of $g$ consists of elements which are zero on the last coordinate (at least), but I'm circling around this with no success.

This is an exercise from Atiyah-Macdonald.

Elinor Rautner
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  • I think an idea which is nice, and is related to you approach using Cayley-Hamilton, is to use exterior powers. Namely, one can show that if $f:M\to N$ is an $R$-linear map between (finitely generated) free modules, then $\wedge^k f:\wedge^k M\to\wedge^k N$ is injective for all $k\geqslant 1$. Applying this to the rank $m$ of $M$ implies that we get an injective map from $\wedge^m M\cong R$ to $\wedge^k N$. If the rank of $N$ was less than m$, then $\wedge^k N=0$ which is a contradiction. – Alex Youcis Feb 20 '16 at 11:19

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