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I am looking for a proof of the following fact: If $R$ is a principal artin local ring and $M$ a finitely generated $R$-module, then $M$ is a direct sum of cyclic $R$-modules. (Apparently such rings $ …

The second result you're talking about is also sometimes called the Chinese remainder theorem, and can be derived from the Chinese remainder theorem for rings by "tensoring the CRT isomorphism" with $ …

I think this is possible, if I understand your question correctly. If $R$ is a discrete valuation ring of mixed characteristic $(0,p)$ with residue field $k$ and maximal ideal $pR$, then $R$ is a Cohe …

Let $\Lambda=\mathbf{Z}_p[[T]]$. The group $M[p]$ is dual to $M^\vee/pM^\vee$. So if $M[p]$ is finite, then $M^\vee/pM^\vee$ is finite, and the form of Nakayama's lemma applicable to profinite modules …

This follows from the Nullstellensatz (or the version of it), which says that a finitely generated algebra over a Jacobson ring is Jacobson. This can be found in, for example, Eisenbud's book on commu …