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Let $R = \mathbb{Z}[i]/(5)$.

It is obvious that $R$ is not an integral domain, and any ideal in $R$ is principal.

Now I want to prove the following classification theorem for modules over $R$ :

There exist modules $M_1, M_2$ such that any finitely generated module $M$ over $R$ is isomorphic to the direct sum $ M_1^r ⊕ M_2^s$, where $M_1^r$ is the direct sum of r copies of module $M_1$, and similarly for $M_2$ .

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    I've removed [tag:algebra] tag, since we don't use algebra tag anymore, see [meta](http://meta.math.stackexchange.com/questions/473/the-use-of-the-algebra-tag/3081#3081) for details.2012-12-14

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