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.

2025年1月20日 · A ring is said to be Noetherian if it is both left and right Noetherian. For a ring R, the following are equivalent: 1. R satisfies the ascending chain condition on ideals (i.e., is Noetherian).

2024年4月3日 · Noetherian rings are named after E. Noether, who made a systematic study of such rings and carried over to them a number of results known earlier only under more stringent restrictions (for example, Lasker's theory of primary decompositions).

A commutative ring R is called Noetherian if each ideal in R is nitely generated. This name honors Emmy Noether, who in her landmark paper [6] in 1921 proved proper-

ring is if every ideal in is finitely generated. In other words, every quotient can be obtained by imposing a finite number of relations. ring is Noetherian if and only if every submodule in a finitely generated -module is itself finitely generated. If is Noetherian, then every finitely generated module is finitely presented.

Noetherian rings have primary decompositions, and simplify the First Uniqueness Theorem concerning the uniqueness of associated prime ideals. Call an ideal I of a ring A irreducible if, for all ideals J , K of A , I = J ∩K =⇒ I = J or I = K. Lemma: Every ideal of a Noetherian ring is a finite intersection of irreducible ideals. 808

10.31 Noetherian rings A ring $R$ is Noetherian if any ideal of $R$ is finitely generated. This is clearly equivalent to the ascending chain condition for ideals of $R$.

If A is a Noetherian ring and f : A → B makes B an A-algebra so that B is a finitely generated A-module under the multiplication a.b = f(a)b, then B is a Noetherian ring.

There are many examples of Noetherian rings. Recall that a ring is said to be a principal ideal domain if every ideal in is a principal ideal, and an ideal is said to be a principal ideal if for some element . So by Theorem 1, we see that every principal ideal domain is a Noetherian ring.

every proper ideal in a Noetherian ring can be expressed as an irredundant intersection of finitely many primary ideals and that any two of such decompositions yield the same set of radical ideals.