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I need an example of a module that is finitely generated, finitely cogenerated and also linearly compact (in discrete topology) but not Artinian.

In fact I proved a theorem with this strong assumptions and I am not sure that there is such a module except finitely generated Artinian modules (and my result for finitely generated Artinian modules is obvious). For the definition of finitely cogenerated modules one can see https://en.wikipedia.org/wiki/Finitely_generated_module. Can anyone give me such an example. Thanks a lot.

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closed as off-topic by user26857, user99914, JonMark Perry, TheSimpliFire, user223391 Feb 4 '18 at 2:36

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    $\begingroup$ Now that we have your "needs" out of the way, perhaps we can briefly look at what you've considered, and how you came about the question? $\endgroup$ – rschwieb Feb 2 '18 at 17:20
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    $\begingroup$ @rschwieb Ok, i proof a theorem with this assumptions: Let $M$ be an $R$-module that is finitely generated, finitely cogenerated and linearly compact! I am not sure that is there any module with this strong assumptions except finitely generated Artinian module. $\endgroup$ – S Ali Mousavi Feb 2 '18 at 17:30
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If you take a Noetherian commutative local ring $R$, complete in the $\mathfrak{m}$-adic topology ($\mathfrak{m}$ the maximal ideal and $E$ the injective envelope of $R/\mathfrak{m}$, then the canonical embedding $R\to\operatorname{End}(E_R)$ is an isomorphism (Matlis, 1958). The trivial extension of $E$ by $R$, that is the ring $A=R\times E$ with operations $$ (r,x)+(s,y)=(r+s,x+y),\qquad (r,x)(s,y)=(rs,ry+xs) $$ is then a ring with a Morita duality induced by the bimodule $_AA_A$ (Dikranjan, Gregorio and Orsatti, 1991, Example 1.13). In particular it is finitely cogenerated and also linearly compact in the discrete topology (Müller, 1970).

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There are three examples in this search of rings which, considered as modules over themselves, have that property.

The easiest one to re-describe here is the trivial extension of the Prufer group $\mathbb Z(p^\infty)$ by $\mathbb Z$. (Its label there is "Finitely cogenerated, not semilocal ring".)

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  • $\begingroup$ Thanks a lot. I saw these examples. Are these also linearly compact? Of course, i know $F[|x|]$ is linearly compact, but what about the field of fractions of this ring? $\endgroup$ – S Ali Mousavi Feb 2 '18 at 17:53
  • $\begingroup$ @SAliMousavi Sorry, I'm not familiar enough with linearly compact modules. I also don't know what you mean by $F[|x|]$: is that your notation for $F[[x]]$? $\endgroup$ – rschwieb Feb 2 '18 at 19:31
  • $\begingroup$ Excuse me. Yes. your notation is correct. $F[[x]]$ $\endgroup$ – S Ali Mousavi Feb 2 '18 at 20:43

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