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 Mα for ordinals α by M0 = 0, Mα+1/Mα = socle M/Mα, 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 epimorphism, where , the socle of is essential in .

Note that if is an artinian module then is a semiartinian module. Clearly 0 is semiartinian.

Let be exact then and are semiartinian if and only if is semiartinian.

Consider 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 left artinian.


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