# 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.

## 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, 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

${\displaystyle _{R}M}$ is a semiartinian module if, for all ${\displaystyle M\rightarrow N}$ epimorphism, where ${\displaystyle N\neq 0}$ , the socle of ${\displaystyle N}$ is essential in ${\displaystyle N}$ .

Note that if ${\displaystyle _{R}M}$ is an artinian module then ${\displaystyle _{R}M}$ is a semiartinian module. Clearly 0 is semiartinian.

Let ${\displaystyle 0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0}$ be exact then ${\displaystyle M'}$ and ${\displaystyle M''}$ are semiartinian if and only if ${\displaystyle M}$ is semiartinian.

Consider ${\displaystyle \{M_{i}\}_{i\in I}}$ family of ${\displaystyle R}$ -modules, then ${\displaystyle \oplus _{i\in I}M_{i}}$ is semiartinian if and only if ${\displaystyle M_{j}}$ is semiartinian for all ${\displaystyle j\in I}$ .

## Semiartinian rings

${\displaystyle R}$ is called left semiartinian if ${\displaystyle _{R}R}$ is semiartinian, that is, ${\displaystyle R}$ is left semiartinian if for any left ideal ${\displaystyle I}$ , ${\displaystyle R/I}$ contains a simple submodule.

Note that ${\displaystyle R}$ left semiartinian does not imply ${\displaystyle R}$ 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, Zbl 1092.16001
• 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, 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, GAUTHIER-VILLARS/EDITIONS ELSEVIER 23 RUE LINOIS, 75015 PARIS, FRANCE, 262: A1295–A1297