# Noetherian module

From Citizendium

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:

- 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 . - Every submodule of
*M*is finitely generated. - 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.