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I need to find all additive endomorphisms of ${\Bbb{Z}}$.

I checked that given an integer $m$, the function $f:{\Bbb{Z}}\to{\Bbb{Z}}$ defined by $f(x) = mx$ is an additive endomorphism. I suspect that every such endomorphism has the form above but I don't know how to prove it.

So, how can I do it?

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1 Answer 1

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Show that $f$ is uniquely determined by $f(1)$. Hint: $f(1+\ldots+1)=f(1)+\ldots +f(1)$.

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