Formally, the ring *R* is **left-Noetherian** iff one (and therefore all) of the following equivalent conditions hold:

- Every left ideal
*I*in*R*is finitely generated, i.e. there exists elements*a*_{1},...,*a*_{n}in*I*such that*I*=*Ra*_{1}+ ... +*Ra*_{n}. - Any ascending chain
*I*_{1}⊆*I*_{2}⊆*I*_{3}⊆ ... of left ideals in*R*eventually becomes stationary: there exists a natural number*n*such that*I*_{m}=*I*_{n}for all*m*≥*n*. This can be rephrased as "the poset of (two-sided) ideals in*R*under inclusion has the ascending chain condition". - Any non-empty set of left ideals of
*R*has a maximal element with respect to set inclusion.

Every field *F* is trivially Noetherian, since it has only two ideals - *F* and {0}. Every finite ring is Noetherian. Other familar examples of Noetherian rings are the ring of integers, **Z**; and **Z**[*x*], the ring of polynomials over the integers. In fact, the Hilbert basis theorem states that if a ring *R* is Noetherian, then the polynomial ring *R*[*x*] is Noetherian as well. If *R* is a Noetherian ring and *I* is an ideal, then the quotient ring *R*/*I* is also Noetherian.
Every commutative Artinian ring is Noetherian.

An example of a ring that's not Noetherian is a ring of polynomials in infinitely many variables: the ideal generated by these variables cannot be finitely generated.

The ring *R* is left-Noetherian if and only if every finitely generated left *R*-module is a Noetherian module.