Noetherian module

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 $M_{0}\subsetneq M_{1}\subsetneq M_{2}\subsetneq \ldots$.
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 finite-dimensional vector space over a field is a Northerian module.

• A free module of infinite rank over an infinite set is not Noetherian.

References

• Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley. ISBN 0-201-55540-9.