Well I am solving old exam tests and I am confused. It is asking to find two non-isomorfic $\mathbb{Z}[i]$-modules with 101 elements each, but I think this can't happen because as abelian groups both will be isomorfic to $\mathbb{Z}_{101}$. It also refers to cyclic $\mathbb{Z}[i]$-torsion modules, but $\mathbb{Z}[i]$ is a PID, so every torsion module isn't free (as a cyclic module would be). Where am I wrong?
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1You may find this link helpful https://math.stackexchange.com/questions/1405419/isomorphism-classes-of-mathbbzi-modules – Math137 Jan 12 '22 at 09:55
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Thanks! This clears things up a bit, but where am I wrong, because I can't see the answer's mistake? Is it because we are talking about module isomorfism and not group isomorfism? @Math137 – Νικολέτα Σεβαστού Jan 12 '22 at 10:18
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1Yes, you need to have the action in mind. Notice that an (a left ) $R$-module $(M, \rho)$ is an abelian group $M$ with a map $\rho: R\times M\to M$ satisfying certain axioms, not just an abelian group $M.$ – Math137 Jan 12 '22 at 12:39
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@ArcticChar Only the first part of the question. – Νικολέτα Σεβαστού Jan 18 '22 at 07:14