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, 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.

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

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