A Noetherian module is a module that satisfies the ascending chain condition on its submodules, i.e., every increasing sequence of submodules eventually becomes constant.

For a unit ring and a left unital R-module , the following are equivalent (Altman and Kleiman 2013, p. 98).

1. satisfies the ascending chain condition on submodules.

2. Every submodule of is a finitely generated module.

3. Every nonempty set of submodules of contains a maximal element with respect to inclusion.

See also

Ascending Chain Condition, Finitely Generated Module, Module, Noetherian Ring

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References

Altman, A. and Kleiman, S. A Term of Commutative Algebra. Version of Sep. 1, 2013. https://web.mit.edu/18.705/www/13Ed.pdf.

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Noetherian Module

Cite this as:

Weisstein, Eric W. "Noetherian Module." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NoetherianModule.html

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