A (left or right) module $M$ over a ring $R$ is said to be Noetherian if the following equivalent conditions hold:

1. Every submodule of $M$ is finitely generated over $R$ .
2. The ascending chain condition holds on submodules.
3. Every nonempty family of submodules has a maximal element.

For example, the $\mathbb{Z}$ -module $\mathbb{Q}$ is not Noetherian, as it is not finitely generated, but the $\mathbb{Z}$ -module $\mathbb{Z}$ is Noetherian, as every submodule is generated by a single element.

Observe that changing the ring can change whether a module is Noetherian or not: for example, the $\mathbb{Q}$ -module $\mathbb{Q}$ is Noetherian, since it is simple (has no nontrivial submodules).

There is also a notion of Noetherian for rings: a ring is left Noetherian if it is Noetherian as a left module over itself, and right Noetherian if it is Noetherian as a right module over itself. For non-commutative rings, these two notions can differ.

The corresponding property for groups is usually called the maximal condition.

Finally, there is the somewhat related notion of a Noetherian topological space.