Length of a module

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In abstract algebra, the length of a module is a measure of the module's "size". It is defined as the length of the longest ascending chain of submodules and is a generalization of the concept of dimension for vector spaces. The 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.

[edit] Definition

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

$N_0\sub N_1 \sub \cdots \sub N_n$

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.

[edit] 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/nZ (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.

[edit] Facts

A module M has finite length if and only if it is both Artinian and Noetherian.

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 M₁ and M₂ have finite length, then so does their direct sum, and the length of the direct sum equals the sum of the lengths of M₁ and M₂.

Suppose

$0\rarr L \rarr M \rarr N \rarr 0$