equivalent defining conditions on a Noetherian ring

Let R be a ring. Then the following are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath:

  1. 1.
  2. 2.

    the ascending chain conditionMathworldPlanetmathPlanetmathPlanetmath on left ideals holds in R,

  3. 3.

    every non-empty family of left ideals has a maximal elementMathworldPlanetmath.


(12). Let I1I2 be an ascending chain of left ideals in R. Let I be the union of all Ij, j=1,2,. Then I is a left ideal, and hence finitely generated, by, say, a1,an. Now each ai belongs to some Iαi. Take the largest of these, say Iαk. Then aiIαk for all i=1,,n, and therefore IIαk. But IαkI by the definition of I, the equality follows.

(23). Let 𝒮 be a non-empty family of left ideals in R. Since 𝒮 is non-empty, take any left ideal I1𝒮. If I1 is maximal, then we are done. If not, 𝒮-{I1} must be non-empty, such that pick I2 from this collectionMathworldPlanetmath so that I1I2 (we can find such I2, for otherwise I1 would be maximal). If I2 is not maximal, pick I3 from 𝒮-{I1,I2} such that I1I2I3, and so on. By assumptionPlanetmathPlanetmath, this can not go on indefinitely. So for some positive integer n, we have In=Im for all mn, and In is our desired maximal element.

(31). Let I be a left ideal in R. Let 𝒮 be the family of all finitely generated ideals of R contained in I. 𝒮 is non-empty since (0) is in it. By assumption 𝒮 has a maximal element J. If JI, then take an element aI-J. Then J,a is finitely generated and contained in I, so an element of 𝒮, contradicting the maximality of J. Hence J=I, in other words, I is finitely generated. ∎

A ring satisfying any, and hence all three, of the above conditions is defined to be a left Noetherian ring. A right Noetherian ring is similarly defined.

Title equivalent defining conditions on a Noetherian ring
Canonical name EquivalentDefiningConditionsOnANoetherianRing
Date of creation 2013-03-22 18:04:27
Last modified on 2013-03-22 18:04:27
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 6
Author CWoo (3771)
Entry type Derivation
Classification msc 16P40