Loewy series

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.

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 M_α for ordinals α by M₀ = 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.

${\displaystyle _{R}M}$ is a semiartinian module if, for all ${\displaystyle M\rightarrow N}$ epimorphism, where ${\displaystyle N\neq 0}$, the socle of ${\displaystyle N}$ is essential in ${\displaystyle N}$.

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