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.
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.
is a semiartinian module if, for all epimorphisms
, the socle of
is essential in
is an artinian module then
is a semiartinian module.
is called left semiartinian if
is left semiartinian if for any left ideal
contains a simple submodule.
left semiartinian does not imply that
is left artinian.