- For any ascending chain of submodules of , there exists an integer so that (i.e. the chain eventually terminates).
Theorem. The following conditions are equivalent for a left -module:
- is noetherian.
- Every submodule of is finitely generated (i.e. can be written as for some ).
- For every collection of submodules of , there is a maximal element.
(The second condition is also frequently used as the definition for noetherian.)
We also have right noetherian modules with the appropriate adjustments.
We say that a ring is left (right) noetherian if it is noetherian as a left (right) -module. If is both left and right noetherian, we call it simply noetherian.
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