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Journal of the London Mathematical Society 1997 55(2):231-235; doi:10.1112/S0024610797004912
© 1997 by London Mathematical Society
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© The London Mathematical Society

Cyclic Artinian Modules without a Composition Series

P. M. Cohn

Department of Mathematics, University College London Gower Street, London WC1E 6BT

Received 21 December 1995. Revision received 27 November 1996.

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.


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