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
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
Every finite-length module M has a composition series, and the length of every such composition series is equal to the length of M.
Read more about this topic: Length Of A Module
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