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How can I prove a well-known fact: $\mathbb {Z}_n (=:\mathbb {Z}/n\mathbb {Z})$ is an injective $\mathbb {Z}_n $-module?

I classified all ideals of $\mathbb {Z}_n $ and tried to use Baer's criterion from the fact that any $\mathbb {Z}_n $-module is a quotient of free one. Is it valid?

user26857
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user253816
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  • Could you provide a few more details? A more straightforward approach would be to take a map $\phi: I\to \mathbb{Z}/n\mathbb{Z}$ for an ideal $I\subset \mathbb{Z}/n\mathbb{Z}$ and extend it to $\hat{\phi}:\mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$. – Andrew Jul 12 '15 at 23:48
  • Oh thanks I misunderstood Baer criterion. It was a stupid question.. haha. – user253816 Jul 13 '15 at 00:43

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