Krull-Remak-Schmidt fails for artinian modules over local rings

Ringel CM (2001)
Algebras and Representation Theory 4(1): 77-86.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
Let R be a ring. Any R-module M which is Artinian or Noetherian can be written as the direct sum of a finite number of indecomposable R-modules. The theorem of Krull-Remak-Schmidt asserts that in the case where M is of finite length, such a decomposition is unique up to isomorphism. On the other hand, examples of Noetherian R-modules which have essentially different decompositions have been known for a long time. The first examples of Artinian R-modules with essentially different decompositions were published only in 1995 by Facchini, Herbera, Levy and Vdmos. In order to construct such examples. one needs to deal with suitable rings R. Note that for R Noetherian or commutative, all the Artinian modules have the Krull-Remak-Schmidt property, In 1998, Facchini raised the problem of whether the same is true in the case where R is a local ring. The aim of this note is to show that this is not so: we are going to present a local ring R and Artinian R-modules M with essentially different direct decompositions into indecomposables. The military importance of these results has been discussed during the NATO meeting at Constantia (August 2000) which was organized by K. W. Roggenkamp.
Stichworte
direct decompositions of modules; Krull-Remak-Schmidt property; Artinian modules; torsion-free Abelian; trivial extension of rings; indecomposable modules; local rings; groups of finite rank
Erscheinungsjahr
2001
Zeitschriftentitel
Algebras and Representation Theory
Band
4
Ausgabe
1
Seite(n)
77-86
ISSN
1386-923X
Page URI
https://pub.uni-bielefeld.de/record/1616014

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Ringel CM. Krull-Remak-Schmidt fails for artinian modules over local rings. Algebras and Representation Theory. 2001;4(1):77-86.
Ringel, C. M. (2001). Krull-Remak-Schmidt fails for artinian modules over local rings. Algebras and Representation Theory, 4(1), 77-86. doi:10.1023/A:1009905728560
Ringel, C. M. (2001). Krull-Remak-Schmidt fails for artinian modules over local rings. Algebras and Representation Theory 4, 77-86.
Ringel, C.M., 2001. Krull-Remak-Schmidt fails for artinian modules over local rings. Algebras and Representation Theory, 4(1), p 77-86.
C.M. Ringel, “Krull-Remak-Schmidt fails for artinian modules over local rings”, Algebras and Representation Theory, vol. 4, 2001, pp. 77-86.
Ringel, C.M.: Krull-Remak-Schmidt fails for artinian modules over local rings. Algebras and Representation Theory. 4, 77-86 (2001).
Ringel, Claus Michael. “Krull-Remak-Schmidt fails for artinian modules over local rings”. Algebras and Representation Theory 4.1 (2001): 77-86.