composition series
Let be a ring and let be a (right or left) -module. A series of submodules
in which each quotient is simple is called a composition series for .
A module need not have a composition series. For example, the ring of integers, , 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 always have composition series.
Title | composition series |
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Canonical name | CompositionSeries |
Date of creation | 2013-03-22 14:04:13 |
Last modified on | 2013-03-22 14:04:13 |
Owner | mclase (549) |
Last modified by | mclase (549) |
Numerical id | 6 |
Author | mclase (549) |
Entry type | Definition |
Classification | msc 16D10 |