Example of a module that is finitely generated, finitely cogenerated and linearly compact, but not Artinian! [closed]

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.

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

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• 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? – rschwieb Feb 2 '18 at 17:20
• @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. – S Ali Mousavi Feb 2 '18 at 17:30

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).

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".)

• 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? – S Ali Mousavi Feb 2 '18 at 17:53
• @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 $F[[x]]$? – rschwieb Feb 2 '18 at 19:31
• Excuse me. Yes. your notation is correct. $F[[x]]$ – S Ali Mousavi Feb 2 '18 at 20:43