# Length of a module

In abstract algebra, the **length** of a module is a measure of the module's "size". It is defined to be the length of the longest chain of submodules and is a generalization of the concept of dimension for vector spaces. Modules with *finite* length share many important properties with finite-dimensional vector spaces.

Other concepts used to 'count' in ring and module theory are depth and height; these are both somewhat more subtle to define. There are also various ideas of *dimension* that are useful. Finite length commutative rings play an essential role in functorial treatments of formal algebraic geometry.

## Definition

Let *M* be a (left or right) module over some ring *R*. Given a chain of submodules of *M* of the form

we say that *n* is the *length* of the chain. The length of *M* is defined to be the largest length of any of its chains. If no such largest length exists, we say that *M* has infinite length.

A ring *R* is said to have finite length as a ring if it has finite length as left *R* module.

## Examples

The zero module is the only one with length 0. Modules with length 1 are precisely the simple modules.

For every finite-dimensional vector space (viewed as a module over the base field), the length and the dimension coincide.

The length of the cyclic group **Z**/*n***Z** (viewed as a module over the integers **Z**)
is equal to the number of prime factors of *n*, with multiple prime factors counted multiple times.

## Facts

A module *M* has finite length if and only if it is both Artinian and Noetherian. (cf. Hopkins' theorem)

If *M* has finite length and *N* is a submodule of *M*, then *N* has finite length as well, and we have length(*N*) ≤ length(*M*). Furthermore, if *N* is a *proper* submodule of *M* (i.e. if it is unequal to *M*), then length(*N*) < length(*M*).

If the modules *M*_{1} and *M*_{2} have finite length, then so does their direct sum, and the length of the direct sum equals the sum of the lengths of *M*_{1} and *M*_{2}.

Suppose

is a short exact sequence of *R*-modules. Then *M* has finite length if and only if *L* and *N* have finite length, and we have

- length(
*M*) = length(*L*) + length(*N*).

(This statement implies the two previous ones.)

A composition series of the module *M* is a chain of the form

such that

A module *M* has finite length if and only if it has a (finite) composition series, and the length of every such composition series is equal to the length of *M*.