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I'm looking for an example for a left R-module that doesn't have the lifting property, From theorems I read, $Z/2Z$ as a $Z$ module should be an example since it' not a direct sum (there isn't a sub-module $K$ of $Z$ such that the direct sum of $Z/2Z$ and $K$ is $Z$).

But I don't understand why $Z/2Z$ doesn't have the lifting property, can someone explain why?

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Suppose $Z/2Z$ had the lifting property. Then in particular the identity map would have a lift, i.e. there would exist a module homomorphism $\varphi: Z/2Z\to Z$ with $\pi\circ\varphi=id$. Show that this yields a contradiction.

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the identity map from Z/2Z to Z is not a homomorphism ...I don't understand what you did –  Belgi Mar 16 '12 at 23:06
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The identity map $Z/2Z$ to $Z/2Z$ would have to have a lift. The epimorphism is $\pi: Z\to Z/2Z$. –  Julian Kuelshammer Mar 16 '12 at 23:08

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