Loewy series

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In mathematics, a left (right) Loewy ring or left (right) semi-Artinian ring is a ring in which every non-zero left (right) module has a non-zero socle, or equivalently if the Loewy length of every left (right) 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), 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

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

Note that if R M {\displaystyle {}_{R}M} {\displaystyle {}_{R}M} is an artinian module then R M {\displaystyle {}_{R}M} {\displaystyle {}_{R}M} is a semiartinian module. Clearly 0 is semiartinian.

If 0 → M ′ → M → M ″ → 0 {\displaystyle 0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0} {\displaystyle 0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0} is exact then M ′ {\displaystyle M'} {\displaystyle M'} and M ″ {\displaystyle M''} {\displaystyle M''} are semiartinian if and only if M {\displaystyle M} {\displaystyle M} is semiartinian.

If { M i } i ∈ I {\displaystyle \{M_{i}\}_{i\in I}} {\displaystyle \{M_{i}\}_{i\in I}} is a family of R {\displaystyle R} {\displaystyle R}-modules, then ⊕ i ∈ I M i {\displaystyle \oplus _{i\in I}M_{i}} {\displaystyle \oplus _{i\in I}M_{i}} is semiartinian if and only if M j {\displaystyle M_{j}} {\displaystyle M_{j}} is semiartinian for all j ∈ I . {\displaystyle j\in I.} {\displaystyle j\in I.}

Semiartinian rings

R {\displaystyle R} {\displaystyle R} is called left semiartinian if R R {\displaystyle _{R}R} {\displaystyle _{R}R} is semiartinian, that is, R {\displaystyle R} {\displaystyle R} is left semiartinian if for any left ideal I {\displaystyle I} {\displaystyle I}, R / I {\displaystyle R/I} {\displaystyle R/I} contains a simple submodule.

Note that R {\displaystyle R} {\displaystyle R} left semiartinian does not imply that R {\displaystyle R} {\displaystyle R} is left artinian.

References