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Could someone suggest a simple $\phi\in $End$_R(A)$ where $A$ is a finitely generated module over ring $R$ where $\phi$ is injective but not surjective? I have a hunch that it exists but I can't construct an explicit example. Thanks.

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    It doesn't have to exist for every ring and every module.2012-03-16
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    They also had a hunch back in Victor Hugo's time: it is to take $R=A=\mathbb Z$ and $\phi(z)=2z$.2012-03-16
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    I am very sorry, I have forgotten to include the condition that $A$ has to be finitely generated.2012-03-16
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    But Georges' $A$ is.2012-03-16
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    Don't worry, Teenager, it was quasi modo implicit.2012-03-16
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    @GeorgesElencwajg: Thanks!2012-03-16
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    @ymar: indeed :) I just thought I should point out my edit2012-03-16
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    Dear @Georges: $+1$ for your mathematical comment, and $+$ the power of the continuum for your puns!2012-03-16
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    Dear @Pierre-Yves, thanks a lot: I really appreciate your kind comment.2012-03-16

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