PlanetMath: composition series

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composition series

(Definition)

Let be a ring and let be a (right or left) -module. A series of submodules

$\displaystyle M = M_0 \supset M_1 \supset M_2 \supset \dots \supset M_n = 0$

in which each quotient $M_i/M_{i+1}$ is simple is called a composition series for

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 above) is called the composition length of the module.

If is a semisimple Artinian ring, then and ${}_RR$ always have composition series.

"composition series" is owned by mclase.

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Cross-references: semisimple, composition, length, artinian, Noetherian, necessary and sufficient, ring of integers, module, simple, quotient, submodules, series, right, ring

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 2018 times total.

Classification:

AMS MSC:

16D10 (Associative rings and algebras :: Modules, bimodules and ideals :: General module theory)

Pending Errata and Addenda

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