length of a module
Let be a ring and let be an -module. If there is a finite sequence of submodules of
such that each quotient module is simple, then is necessarily unique by the Jordan-Hölder theorem (http://planetmath.org/JordanHolderDecomposition) for modules. We define the above number to be the length of . If such a finite sequence does not exist, then the length of is defined to be .
If has finite length, then satisfies both the ascending and descending chain conditions.
A ring is said to have finite length if there is an -module whose length is finite.
Title | length of a module |
---|---|
Canonical name | LengthOfAModule |
Date of creation | 2013-03-22 14:35:32 |
Last modified on | 2013-03-22 14:35:32 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 11 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 16D10 |
Classification | msc 13C15 |
Synonym | finite-length module |
Defines | finite length |