In mathematics, a left (right) Loewy ring or left (right) semi-Artinian ring is a ring in which every non-zero left (right) module has a non-zero socle, or equivalently if the Loewy length of every left (right) module is defined. The concepts are named after Alfred Loewy.

Loewy length

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The Loewy length and Loewy series were introduced by Emil Artin, Cecil J. Nesbitt, and Robert M. Thrall (1944).

If M is a module, then define the Loewy series Mα for ordinals α by M0 = 0, Mα+1/Mα = socle(M/Mα), and Mα = ∪λ<α Mλ if α is a limit ordinal. The Loewy length of M is defined to be the smallest α with M = Mα, if it exists.

Semiartinian modules

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  is a semiartinian module if, for all epimorphisms  , where  , the socle of   is essential in  

Note that if   is an artinian module then   is a semiartinian module. Clearly 0 is semiartinian.

If   is exact then   and   are semiartinian if and only if   is semiartinian.

If   is a family of  -modules, then   is semiartinian if and only if   is semiartinian for all  

Semiartinian rings

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  is called left semiartinian if   is semiartinian, that is,   is left semiartinian if for any left ideal  ,   contains a simple submodule.

Note that   left semiartinian does not imply that   is left artinian.

References

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  • Assem, Ibrahim; Simson, Daniel; Skowroński, Andrzej (2006), Elements of the representation theory of associative algebras. Vol. 1: Techniques of representation theory, London Mathematical Society Student Texts, vol. 65, Cambridge: Cambridge University Press, ISBN 0-521-58631-3, Zbl 1092.16001
  • Artin, Emil; Nesbitt, Cecil J.; Thrall, Robert M. (1944), Rings with Minimum Condition, University of Michigan Publications in Mathematics, vol. 1, Ann Arbor, MI: University of Michigan Press, MR 0010543, Zbl 0060.07701
  • Nastasescu, Constantin; Popescu, Nicolae (1968), "Anneaux semi-artiniens", Bulletin de la Société Mathématique de France, 96: 357–368, ISSN 0037-9484, MR 0238887, Zbl 0227.16014
  • Nastasescu, Constantin; Popescu, Nicolae (1966), "Sur la structure des objets de certaines catégories abéliennes", Comptes Rendus de l'Académie des Sciences, Série A, 262, GAUTHIER-VILLARS/EDITIONS ELSEVIER 23 RUE LINOIS, 75015 PARIS, FRANCE: A1295–A1297