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.
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. Both concepts are named after Emmy Noether.
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.
