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

Question

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.

Answer 1

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).

Answer 2

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".)