For $A$-modules and homomorphisms $0\to M'\stackrel{u}{\to}M\stackrel{v}{\to}M''\to 0$ is exact. Prove if $M'$ and $M''$ are fintely generated then $M$ is finitely generated.
Finitely generated modules in exact sequence
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abstract-algebra
modules
exact-sequence
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2Do this for vector spaces. Then do exactly the same for modules. – 2012-11-11
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2See the proof of the lemma of my answer to this question. http://math.stackexchange.com/questions/231058/noetherian-rings-and-modules – 2012-11-11