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Let $A$ a ring, for each monomorphism $f:A^m \rightarrow A^n$, I don't know how to prove that $m\leq n$. I can't start the problem, I have no idea, help me please.

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marked as duplicate by rschwieb, Davide Giraudo, Henry T. Horton, Amzoti, Thomas Mar 29 '13 at 17:18

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Duplicate of – Zev Chonoles Mar 29 '13 at 16:37
Presumably, you mean monomorphisms as $A$-modules? – Thomas Andrews Mar 29 '13 at 16:37
If indeed monomorphisms of $A$ modules, are commutative rings intended? There is a noncommutative ring with $R^n\cong R^m$ as right $R$ modules for every pair of positive integers $m,n$. – rschwieb Mar 29 '13 at 16:41
I have a doubt, if f is a monomorphism then f is injective what means that for each element of $A^m$ has an only image in $A^n$, so if I suggest that m>n then some elements of $A^m$ have the same image in $A^n$. It is correct in this exercise? – rgl4 Apr 7 '13 at 20:46

Hint: Exterior power functors preserve monomorphisms of free modules. Use this to get an inequality involving binomial coefficients.

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