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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.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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