Length (algebra)

In the mathematical branch of algebra , length is a measure of the size of a module .

definition

Let it be a module over a ring . The length of is the supremum of the lengths of chains of sub-modules of form ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle M}$ ${\displaystyle n}$

${\displaystyle 0=N_{0}\subsetneq N_{1}\subsetneq N_{2}\subsetneq \dotsb \subsetneq N_{n}=M.}$

The length is often referred to as or . ${\displaystyle \ell _{A}(M)}$ ${\displaystyle \ell (M)}$

properties

• Only the zero module has length 0.
• A module is simple if and only if its length is 1.
• A module has finite length if and only if it is Artinian and Noetherian .
• The length is additive to short exact sequences : Actual

${\displaystyle 0\to M'\to M\to M''\to 0}$

exactly, so is ; if two of these numbers are finite, so is the third. ${\displaystyle \ell (M)=\ell (M')+\ell (M'')}$

• A composition series is a chain of sub-modules that has simple sub-quotients. The length of each composition series is equal to the length of the module.

Examples

• Vector spaces have finite length if and only if they are finite-dimensional; in this case its length is equal to its dimension.
• The module has infinite length: for every natural number is ${\displaystyle \mathbb {Z} }$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle n}$

${\displaystyle 0\subset 2^{n}\mathbb {Z} \subset 2^{n-1}\mathbb {Z} \subset \dotsb \subset \mathbb {Z} }$

a chain of sub-modules of length . ${\displaystyle n+1}$

literature

• Henning Krause, Claus Michael Ringel ed .: Infinite length modules . Birkhäuser, Basel 2000, ISBN 3-7643-6413-0 .

Individual evidence

1. ^ Siegfried Bosch : Algebra , 6th edition 2006, Springer-Verlag, ISBN 3-540-40388-4 , p. 72.
2. ↑ Henning Krause, Claus Michael Ringel ed .: Infinite length modules . Birkhäuser, Basel 2000, ISBN 3-7643-6413-0 , p. 3 .