PlanetMath: length of a module

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length of a module

(Definition)

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.

"length of a module" is owned by CWoo. [ full author list (2) | owner history (1) ]

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Other names:

finite-length module

Also defines:

finite length

Keywords:

length

Attachments:: Fitting's lemma (Theorem) by CWoo

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Cross-references: finite, descending chain conditions, length, number, modules, simple, quotient module, submodules, finite sequence, ring
There are 7 references to this entry.

This is version 8 of length of a module, born on 2004-09-09, modified 2007-08-20.
Object id is 6156, canonical name is Length.
Accessed 5883 times total.

Classification:

AMS MSC:	13C15 (Commutative rings and algebras :: Theory of modules and ideals :: Dimension theory, depth, related rings )
	16D10 (Associative rings and algebras :: Modules, bimodules and ideals :: General module theory)

Pending Errata and Addenda

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