If M is a module and M is a
submodule of M, then N is irreducible in M if N cannot be written as a proper
intersection of two submodules of M. The purpose of this note is to study modules
whose submodules can be written as a finite intersection of irreducible submodules.
Such modules are characterized by the fact that their quotients all have finite Goldie
dimension, so they are called q.f.d. modules.
The main result is: A module M is q.f.d. if and only if every submodule N has a
finitely generated submodule T such that N∕T has no maximal submodules.
Because T is finitely generated this generalizes a theorem of Shock (using
his ideas), who showed a q.f.d. module M having the property that every
subquotient of M has a maximal submodule must be noetherian (and conversely, of
course).
