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.
An example of an endomorphism
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$\begingroup$
modules
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0It doesn't have to exist for every ring and every module. – 2012-03-16
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4They 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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0I am very sorry, I have forgotten to include the condition that $A$ has to be finitely generated. – 2012-03-16
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2But Georges' $A$ is. – 2012-03-16
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1Don't worry, Teenager, it was quasi modo implicit. – 2012-03-16
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0@GeorgesElencwajg: Thanks! – 2012-03-16
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0@ymar: indeed :) I just thought I should point out my edit – 2012-03-16
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0Dear @Georges: $+1$ for your mathematical comment, and $+$ the power of the continuum for your puns! – 2012-03-16
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0Dear @Pierre-Yves, thanks a lot: I really appreciate your kind comment. – 2012-03-16