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.
The Loewy length and Loewy series were introduced by .
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.
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M → N
N ≠ 0
N
N.
Note that if
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If
0 → M' → M → M'' → 0
M'
M''
M
If
\{Mi\}i\in
R
⊕ i\inMi
Mj
j\inI.
R
RR
R
I
R/I
Note that
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