What is wrong in my proof for "every linearly compact module is Artinian"

Question

We know that every artinian module is linearly compact but the converse is not true. But i can proof the converse!!

By a theorem, a module $M$ is artinian iff every each of quotient modules is finitely cogenerated. (Algebras, Rings and Modules, Volume 1. Michiel Hazewinkel, Nadiya Gubareni, V.V. Kirichenko. Prop. 3.1.6)

On the other hand we know that every linearly compact module is finitely cogenerated. ( Modules over Non-Noetherian Domains, L. Fuchs and Luigi Salce, Lemma 7.3).

Also every quotient module of a linearly compact module is linearly compact.(Foundation of module and rings, robert Wisbauer. 29.8)

Hence, every quotient module of a linearly compact module is finitely cogenerated and proof is complete!

What is wrong in this proof. Please help me. Thanks a lot.

Here is Lemma 7.3 page 298 of L. Fuch' book.

Lemma 7.3. A module that is linearly compact in the discrete topology is finitely cogenerated, so it has finite Goldie dimension.It also has finite dual Goldie dimension.

Answer 1

As confirmed in an email from the book's second author, the statement of Lemma 7.3 is incorrect; it should only say that a linearly compact module (with the discrete topology) has finite Goldie dimension and codimension. (This part is correct.)

There are several counterexamples to the statement that a linearly compact module is finitely cogenerated. The simplest one is the ring $F[[x]]$ of formal power series over the field $F$, which is a complete local ring, hence linearly compact in every linear topology, including the discrete one, by a result of Zelinsky: Linearly Compact Modules and Rings, American Journal of Mathematics, Vol. 75, No. 1 (Jan., 1953), pp. 79–90.

Note that $F[[x]]$, as a module over itself, is not finitely cogenerated, having zero socle.

Answer 2

No, you misread the conditions of Fuchs' lemma $7.3$ (page 298). It doesn't say that any linearly compact module is finitely cogenerated. What the lemma says is that if a module is linearly compact in the discrete topology, then it's finitely cogenerated.

Thus, what you've proved is that a module is artinian if and only if it's linearly compact in the discrete topology.