Noetherian module
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In abstract algebra, an Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion.
Historically, Hilbert was the first mathematician to work with the properties of finitely generated submodules. He proved an important theorem known as Hilbert's basis theorem which says that any ideal in the multivariate polynomial ring of an arbitrary field is finitely generated. However, the property is named after Emmy Noether who was the first one to discover the true importance of the property.
Two other equivalent conditions are: a module is Noetherian if and only if all of its submodules are finitely generated, if and only if any nonempty set S of submodules has a maximal element (by inclusion).
A Noetherian ring is, by definition, a Noetherian module over itself. Any finitely generated module over a Noetherian ring is a Noetherian module.
If M is a module and K a submodule, then M is Noetherian if and only if K and M/K are Noetherian. This is in contrast to the general situation with finitely generated modules: a submodule of a finitely generated module need not be finitely generated.
[edit] See also
[edit] References
- Eisenbud Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag, 1995.
