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 (Atiyah and Macdonald 1969, pp. 76-77).

A module has finite length iff it is both Artinian and Noetherian; this includes the case where is finite (Atiyah and Macdonald 1969, p. 77).

An abstract vector space has finite length iff it is finite-dimensional, and in this case the length coincides with the dimension (Atiyah and Macdonald 1969, p. 78).

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.

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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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