# 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

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

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

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

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

## Semiartinian rings

$R$ is called left semiartinian if $_{R}R$ 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 $R$ left artinian.

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