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 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

Ring theory