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Homelength of a module

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# length of a module

Let $A$ be a ring and let $M$ be an $A$-module. If there is a finite sequence of submodules of $M$

$\displaystyle M=M_{0}\supset M_{1}\supset\cdots\supset M_{n}=0$ |

such that each quotient module $M_{i}/M_{{i+1}}$ is simple, then $n$ is necessarily unique by the Jordan-Hölder theorem for modules. We define the above number $n$ to be the *length* of $M$. If such a finite sequence does not exist, then the length of $M$ is defined to be $\infty$.

If $M$ has finite length, then $M$ satisfies both the ascending and descending chain conditions.

A ring $A$ is said to have *finite length* if there is an $A$-module whose length is finite.

Defines:

finite length

Keywords:

length

Synonym:

finite-length module

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

16D10*no label found*13C15

*no label found*

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