Loewy ring

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 M_α for ordinals α by M₀ = 0, M_α+1/M_α = socle(M/M_α), and M_α = ∪_λ<α M_λ if α is a limit ordinal. The Loewy length of M is defined to be the smallest α with M = M_α, if it exists.

${\displaystyle {}_{R}M}$ is a semiartinian module if, for all epimorphisms ${\displaystyle M\rightarrow N}$, 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.

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

If ${\displaystyle \{M_{i}\}_{i\in I}}$ is a 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 ${\displaystyle j\in I.}$

${\displaystyle R}$ 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 that ${\displaystyle R}$ is left artinian.

• 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
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