Noetherian ring

In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence of left (or right) ideals has a largest element; that is, there exists an n such that: Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems on rings rely heavily on Noetherian property (for example, the Lasker–Noether theorem and the Krull intersection theorem). Noetherian rings are named after Emmy Noether, but the importance of the concept was recognized earlier by David Hilbert, with the proof of Hilbert's basis theorem (which asserts that polynomial rings are Noetherian) and Hilbert's syzygy theorem. For noncommutative rings, it is necessary to distinguish between three very similar concepts: A ring is left-Noetherian if it satisfies the ascending chain condition on left ideals. A ring is right-Noetherian if it satisfies the ascending chain condition on right ideals. A ring is Noetherian if it is both left- and right-Noetherian. For commutative rings, all three concepts coincide, but in general they are different. There are rings that are left-Noetherian and not right-Noetherian, and vice versa. There are other, equivalent, definitions for a ring R to be left-Noetherian: Every left ideal I in R is finitely generated, i.e. there exist elements in I such that . Every non-empty set of left ideals of R, partially ordered by inclusion, has a maximal element. Similar results hold for right-Noetherian rings.