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 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.

Semiartinian modules

[math]\displaystyle{ {}_R M }[/math] is a semiartinian module if, for all epimorphisms [math]\displaystyle{ M \rightarrow N }[/math], where [math]\displaystyle{ N \neq 0 }[/math], the socle of [math]\displaystyle{ N }[/math] is essential in [math]\displaystyle{ N. }[/math]

Note that if [math]\displaystyle{ {}_R M }[/math] is an artinian module then [math]\displaystyle{ {}_R M }[/math] is a semiartinian module. Clearly 0 is semiartinian.

If [math]\displaystyle{ 0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0 }[/math] is exact then [math]\displaystyle{ M' }[/math] and [math]\displaystyle{ M'' }[/math] are semiartinian if and only if [math]\displaystyle{ M }[/math] is semiartinian.

If [math]\displaystyle{ \{M_i\}_{i\in I} }[/math] is a family of [math]\displaystyle{ R }[/math]-modules, then [math]\displaystyle{ \oplus_{i\in I}M_{i} }[/math] is semiartinian if and only if [math]\displaystyle{ M_j }[/math] is semiartinian for all [math]\displaystyle{ j \in I. }[/math]

Semiartinian rings

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

Note that [math]\displaystyle{ R }[/math] left semiartinian does not imply that [math]\displaystyle{ R }[/math] 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