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 
If "M" is a module, then define the Loewy series "M"α for ordinals α by "M"0 = 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.
Semiartinian modules.
formula_1 is a semiartinian module if, for all formula_2 epimorphism, where formula_3, the socle of formula_4 is essential in formula_4.
Note that if formula_1 is an artinian module then formula_1 is a semiartinian module. Clearly 0 is semiartinian.
Let formula_8 be exact then formula_9 and formula_10 are semiartinian if and only if formula_11 is semiartinian.
Consider formula_12 family of formula_13-modules, then formula_14 is semiartinian if and only if formula_15 is semiartinian for all formula_16.
Semiartinian rings.
formula_13 is called left semiartinian if formula_18 is semiartinian, that is, formula_13 is left semiartinian if for any left ideal formula_20, formula_21 contains a simple submodule.
Note that formula_13 left semiartinian does not imply formula_13 left artinian.