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Cyclic Artinian Modules without a Composition Series
Author(s) -
Cohn P. M.
Publication year - 1997
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610797004912
Subject(s) - mathematics , artinian ring , noetherian , semisimple module , pure mathematics , discrete mathematics , algebra over a field , finitely generated abelian group , ring (chemistry) , noncommutative ring , chemistry , organic chemistry
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.