I am studing the book "Rings and Categories of modules" written by Frank W. Anderson and Kent R. Fuller. By definition posted below one can infer that a submodule of a f.c.o module is also f.c.o. However,what about a fractor module of f.c.o module?The answer seems negative but i can't find a example.Can someone give me some hint?
It is not hard to see that a module is Artinian iff every quotient is finitely cogenerated.
So in principle, it is obvious what to aim for: a finitely cogenerated module $M$ that isn't Artinian. There must be a strictly descending chain of submodules with nonzero intersection $N$, and then $M/N$ is not finitely cogenerated.
There exist rings (like this one) which are finitely cogenerated as modules over themselves, but they have quotients which aren't Artinian. In the linked example above, the ring has a quotient isomorphic to $F[[x]]$ for a field $x$, which is a nonArtinian module over the ring. However, the ring itself is finitely cogenerated because its ideals are linearly ordered, and there is a smallest nonzero submodule.