Loewy ring

In mathematics, a left Loewy ring or left semi-Artinian ring is a ring in which every non-zero left module has a non-zero socle, or equivalently if the Loewy length of every left module is defined. The concepts are named after Alfred Loewy.

Loewy length

The Loewy length and Loewy series were introduced by.
If M is a module, then define the Loewy series M_α for ordinals α by M₀ = 0, M_α+1/M_α = socle, 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

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 sequence|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.