Length of a module

In the mathematical field of algebra, the length refers to a measure of the size of a module.

Definition

It should be a module over a ring. The length of the supremum of lengths of chains of submodules of the form

The length is often referred to with or.

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 on short exact sequences: Is
  • A composition series is a chain of sub-modules, which has simple subquotients. 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; in this case, its length equal to its dimension.
  • The module has infinite length: For each natural number
Simple module Siegfried Bosch
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