The module length of a module 
is the length of any of its composition
series. According to the Jordan-Hölder
theorem for modules, if 
has any composition series,
then all such series are equivalent. The length of a module
without composition series is conventionally
set equal to 
.
A module has finite length iff it is both Artinian
and Noetherian; this includes the case where
is finite.
An abstract vector space has finite length iff it is finite-dimensional, and in this case the
length coincides with the dimension.
See also
Composition Series,
Dimension,
Hilbert-Samuel Function,
Hilbert-Samuel
Multiplicity,
Jordan-Hölder Theorem,
Serre's Intersection Multiplicity
This entry contributed by Margherita
Barile
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References
Atiyah, M. F. and Macdonald, I. G. Introduction to Commutative Algebra. Menlo Park, CA: Addison-Wesley, pp. 76-78, 1969.Referenced
on Wolfram|Alpha
Module Length
Cite this as:
Weisstein, Eric W., with contributions by Margherita Barile. "Module Length." From MathWorld--A Wolfram
Resource. https://mathworld.wolfram.com/ModuleLength.html
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