In mathematics, a left (right) Loewy ring or left (right) semi-Artinian ring is a ring in which every non-zero left (right) module has a non-zero socle, or equivalently if the Loewy length of every left (right) 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), 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
is a semiartinian module if, for all epimorphisms , where , the socle of is essential in
Note that if is an artinian module then is a semiartinian module. Clearly 0 is semiartinian.
If is exact then and are semiartinian if and only if is semiartinian.
If is a family of -modules, then is semiartinian if and only if is semiartinian for all
Semiartinian rings
is called left semiartinian if is semiartinian, that is, is left semiartinian if for any left ideal , contains a simple submodule.
Note that left semiartinian does not imply that 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, 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
- Nastasescu, Constantin; Popescu, Nicolae (1966), "Sur la structure des objets de certaines catégories abéliennes", Comptes Rendus de l'Académie des Sciences, Série A, 262, GAUTHIER-VILLARS/EDITIONS ELSEVIER 23 RUE LINOIS, 75015 PARIS, FRANCE: A1295 – A1297
