Noetherian ring

In mathematics, a ring is called Noetherian if, intuitively speaking, it is not "too large" as expressed by a certain finiteness condition on its idealss. Noetherian rings are named after the mathematician Emmy Noether, who developed much of their theory.

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 a₁,...,a_n in I such that I = Ra₁ + ... + Ra_n.
• Any ascending chain I₁ ⊆ I₂ ⊆ I₃ ⊆ ... 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.

The ring R is called right-Noetherian if the above conditions are true for right ideals, and it is called Noetherian if it is both left-Noetherian and right-Noetherian. For commutative rings, the three notions "left-Noetherian", "right-Noetherian" and "Noetherian" coincide.

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.