|
|
|
|
composition series
|
(Definition)
|
|
|
Let $R$ be a ring and let $M$ be a (right or left) $R$ -module. A series of submodules $$M = M_0 \supset M_1 \supset M_2 \supset \dots \supset M_n = 0$$ in which each quotient $M_i/M_{i+1}$ is simple is called a composition series for $M$ .
A module need not have a composition series. For example, the ring of integers, $\mathbb{Z}$ , considered as a module over itself, does not have a composition series.
A necessary and sufficient condition for a module to have a composition series is that it is both Noetherian and Artinian.
If a module does have a composition series, then all composition series are the same length. This length (the number $n$ above) is called the composition length of the module.
If $R$ is a semisimple Artinian ring, then $R_R$ and ${}_RR$ always have composition series.
|
"composition series" is owned by mclase.
|
|
(view preamble | get metadata)
Cross-references: semisimple, composition, number, length, artinian, Noetherian, necessary and sufficient, ring of integers, module, simple, quotient, submodules, series, right, ring
There is 1 reference to this entry.
This is version 3 of composition series, born on 2003-11-22, modified 2006-10-18.
Object id is 5428, canonical name is CompositionSeries3.
Accessed 2701 times total.
Classification:
| AMS MSC: | 16D10 (Associative rings and algebras :: Modules, bimodules and ideals :: General module theory) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|