In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion.
[1] 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.
When R is a commutative ring the left-right adjectives may be dropped as they are unnecessary.
Since a sub-bimodule of an R-S bimodule M is in particular a left R-module, if M considered as a left R-module were Noetherian, then M is automatically a Noetherian bimodule.