ABSTRACT
An interesting property of artinian modules is that they have finitely generated essential socle. The modules with this property are called finitely embedded or finitely cogenerated and, in fact, a module M is artinian if and only if every quotient module of M is finitely cogenerated. Showing that a module (or ring) has this property is often a useful step towards proving that the module is artinian but, as we will indicate below, there are also nonartinian finitely cogenerated rings which are interesting in their own right.
