Cyclic Artinian Modules without a Composition Series

Abstract

Let R be a ring (always understood to be associative with a unit element 1). It is well known that an R-module is Noetherian if and only if all its submodules are finitely generated and that it has a finite composition series if and only if it is Noetherian and Artinian. This raises the question whether every finitely generated Artinian module is Noetherian; here it is enough to consider cyclic Artinian modules, by an induction on the length. This question has been answered (negatively) by Brian Hartley [5], who gives a construction of an Artinian uniserial module of uncountable composition-length over the group algebra of a free group of countable rank. If we are just interested in finding cyclic modules that are Artinian but not Noetherian, there is a very simple construction based on the fact that over a free algebra every countably generated Artinian module can be embedded in a cyclic module which is again Artinian. This is described in §2 below.