Noetherian module

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In algebra, a Noetherian module is a module with a condition on the lattice of submodules.

Definition

Fix a ring R and let M be a module. The following conditions are equivalent:

  1. The module M satisfies an ascending chain condition on the set of its submodules: that is, there is no infinite strictly ascending chain of submodules .
  2. Every submodule of M is finitely generated.
  3. Every nonempty set of submodules of M has a maximal element when considered as a partially ordered set with respect to inclusion.

When the above conditions are satisfied, M is said to be Noetherian.

Examples

  • A zero module is Noetherian, since its only submodule is itself.
  • A Noetherian ring (satisfying ACC for ideals) is a Noetherian module over itself, since the submodules are precisely the ideals.
  • A free module of finite rank over a Noetherian ring is a Noetherian module.
  • A free module of infinite rank over an infinite set is not Noetherian.

References