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I have tried to find a left module $M$ over an artinian ring $R$ with an injective endomorphism that is not an automorphism, so is there any suggestions, please?

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Take $R=F$ to be a field, and consider $M=\prod_{i\in \mathbb N}F$.

You can inject $M$ onto the subspace of $M$ whose even coordinates are $0$, but this map is not surjective.

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  • $\begingroup$ but i need map between V to itself not to subspace of V ! $\endgroup$ – Maram Os Nov 23 '16 at 21:38
  • $\begingroup$ @MaramOs You understand that a subspace of $M$ is a subset of $M$, right? $\endgroup$ – rschwieb Nov 23 '16 at 21:51
  • $\begingroup$ @rschwieb yes, they are subsets and vector spaces of V $\endgroup$ – Maram Os Nov 23 '16 at 22:14
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    $\begingroup$ @MaramOs So a map from $M$ to a subspace of $M$ is an endomorphism of $M$. $\endgroup$ – rschwieb Nov 23 '16 at 22:48
  • $\begingroup$ "In abstract algebra, the endomorphism ring of an abelian group $X$, denoted by $End(X)$, is the set of all homomorphisms of $X$ into itself" $\endgroup$ – Maram Os Nov 24 '16 at 11:59

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