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 right Noetherian ring R is, by definition, a Noetherian right R module over itself using multiplication on the right. Likewise a ring is called left Noetherian ring when R is Noetherian considered as a left R module. When R is a commutative ring the left-right adjectives may be dropped, as they are unnecessary. Also, if R is Noetherian on both sides, it is customary to call it Noetherian and not "left and right Noetherian".
Any finitely generated right module over a right 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.
The Noetherian condition can also be defined on bimodule structures as well: a Noetherian bimodule is a bimodule whose poset of sub-bimodules satisfies the ascending chain condition. Since a sub-bimodule of an R-S bimodule M is a fortiori a left R-module, if M considered as a left R module were Noetherian, then M is automatically a Noetherian bimodule. It may happen, however, that a bimodule is Noetherian without its left or right structures being Noetherian.