a question about artinian modules

Question

Let M be a modules. Are the following equivalent.

i) M is artinian ii) every factor of $M$ is finitely generated.

Yours,

Answer 1

No, the conditions aren't equivalent. The classical example of a non finitely generated artinian module is the Prüfer $p$-group $$ \mathbb{Z}(p^\infty)=\varinjlim \mathbb{Z}/p^n\mathbb{Z} $$ Each of its nontrivial quotient is isomorphic to it, so none of its quotients (except the trivial one) is finitely generated.

As Prahlad Vaidyanathan remarks, $\mathbb{Z}$ is not artinian and all of its quotients are finitely generated. Actually, every finitely generated module would satisfy your second condition, but finitely generated modules are generally not artinian.

The equivalent conditions are

1. The module $M$ is artinian
2. Every quotient module of $M$ is finitely cogenerated

A module $M$ is finitely cogenerated if for each family of submodules of $M$ closed under finite intersections, if the intersection of the family is $\{0\}$ then $\{0\}$ belongs to the family. Equivalently, if $M$ embeds in a product, then this embedding factors through a finite product.

Another equivalent characterization of finitely cogenerated module is that they have finitely generated and essential socle.