Loewy ring

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

## Contents

* 1 Loewy length
* 2 Semiartinian modules
* 3 Semiartinian rings
* 4 References

## Loewy length

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

RM is a semiartinian module if, for all epimorphisms M->N, where N≠0, the socle of N is essential in N.

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

If 0->M′->M->M″->0 is exact then M′ and M″ are semiartinian if and only if M is semiartinian.

If {Mi}i∈I is a family of R-modules, then ⊕i∈IMi is semiartinian if and only if Mj is semiartinian for all j∈I.

## Semiartinian rings

R is called left semiartinian if RR is semiartinian, that is, R is left semiartinian if for any left ideal I, R/I contains a simple submodule.

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

## References

* 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, 65, Cambridge: Cambridge University Press, ISBN 0-521-58631-3
* Artin, Emil; Nesbitt, Cecil J.; Thrall, Robert M. (1944), Rings with Minimum Condition, University of Michigan Publications in Mathematics, 1, Ann Arbor, MI: University of Michigan Press, https://books.google.com/books?id=ZOcyYgEACAAJ
* Nastasescu, Constantin; Popescu, Nicolae (1968), "Anneaux semi-artiniens", Bulletin de la Société Mathématique de France 96: 357–368, ISSN 0037-9484, http://www.numdam.org/item?id=BSMF_1968__96__357_0
* Nastasescu, Constantin; Popescu, Nicolae (1966), "Sur la structure des objets de certaines catégories abéliennes", Comptes Rendus de l'Académie des Sciences, Série A (GAUTHIER-VILLARS/EDITIONS ELSEVIER 23 RUE LINOIS, 75015 PARIS, FRANCE) 262: A1295-A1297

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