Loewy ring

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 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.

Semiartinian modules

R M is a semiartinian module if, for all M → N epimorphism, where N ≠ 0 , the socle of N is essential in N .

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

Let 0 → M ′ → M → M ″ → 0 be exact then M ′ and M ″ are semiartinian if and only if M is semiartinian.

Let us consider { M i } i ∈ I family of R -modules, then ⊕ i ∈ I M i is semiartinian if and only if M j is semiartinian for all j ∈ I .

Semiartinian rings

R is called left semiartinian if R R is semiartinian, that is, R is left semiartinian if for any left ideal I , R / I contains a simple submodule.

Note that R left semiartinian does not imply R left artinian.