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 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.
Definition
Let M be a (left or right) module over some ring R. Given a chain of submodules of M of the form
- <math>N_0\sub N_1 \sub \cdots \sub N_n<math>
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.
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.
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
- <math>0\rarr L \rarr M \rarr N \rarr 0<math>
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
- <math>0=N_0\sub N_1 \sub \cdots \sub N_n=M<math>
such that
- <math>N_{i+1}/N_i \mbox{ is simple for }i=0,\dots,n-1<math>
Every finite-length module M has a composition series, and the length of every such composition series is equal to the length of M.