Aring is called left (respectively, right) Noetherian if it does not contain an infinite ascending chain ofleft (respectively,right) ideals. In this case, thering in question is said to satisfy theascending chain condition onleft (respectively,right) ideals.

Aring is said to be Noetherian if it is both left and right Noetherian. For aring, the following are equivalent:

1. satisfies theascending chain condition onideals (i.e., is Noetherian).

2. Everyideal of isfinitely generated.

3. Every set ofideals contains amaximalelement.

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Cite this as:

Weisstein, Eric W. "Noetherian Ring."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/NoetherianRing.html

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