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Prove that $\mathbb{Q} $ is not a projective $\mathbb{Z} $ module

Let on the contrary it is projective.

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Then $P=\mathbb{Q}$. Then bottom row is given to be exact. Now h is given to be an $\mathbb{Z}$-module homomorphism such that gh=f. But I am not able to think about what contradiction I can obtain.

I am not very good in projective modules.

  • Do you know any other characterizations of projective modules? – Aphelli Oct 13 '21 at 13:56
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    Please search before asking. These were linked in the right hand column as "related", so the search feature should have also shown them when you provided the title of the question. – rschwieb Oct 13 '21 at 15:14

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