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In mathematics finiteness properties are very important. The Noetherian condition (named in honour of the great algebraist Emmy Noether) is such a property. It plays an important rôle in algebra and algebraic geometry. See also the (essentially) dual notion of the Artinian condition.

First of all you will want to read ideal to find out what a right ideal, a left ideal and a two-sided ideal are (and to absorb some notation). But remember for commutative rings these conditions all coincide as do the properties of being right Noetherian, left Noetherian, and Noetherian that I define below. The first time you read this you should probably just think about commutative rings and forget about left and right.

Definition A right ideal I in a ring R is called finitely generated if there exist finitely many elements a1,...,an of R such that

I=a1R +...+ anR
We define the notion of finitely generated for a left ideal similarly.

Definition A ring R is called right Noetherian if every right ideal is finitely generated. R is called left Noetherian if every left ideal is finitely generated. R is called Noetherian if it is both left and right Noetherian.

We could have given two alternative definitions for the Noetherian property as the following result shows.

Theorem Let R be a ring. Then TFAE

  1. R is right Noetherian.
  2. whenever I1 <= I2 <= ... is an ascending chain (here I use "<=" to mean "is a subset of") of right ideals it eventually becomes stationary. i.e. for N suitably large In=IN for all n>N. This is called the ascending chain condition.
  3. Any nonempty set of right ideals of R contains a maximal element. i.e. if S is such a nonempty set there exists a right ideal I in S such that there is no right ideal J in S that contains I and is not equal to I. Note that there can be more than one such maximal element. This is called the maximum condition.
Proof: 1 ==> 2. Let I be the union of all the right ideals In in the chain. I is itself a right ideal, for if a,b are elements of I then, by definition, a is in some In and b is in some Im. So they both lie in Imax(m,n). Clearly then a-b and ar (for r in R) are also in this latter right ideal and hence in I. By 1 then we know that I has finitely many generators, say a1,...,ar. Choose N so that each of these generators is in IN. Then the chain becomes stationary at this point.

2 ==> 1. Let I be a right ideal that is not finitely generated. Suppose that we have chosen a1,...,an in I. Then, by assumption, I is not equal to a1R +...+ anR. So we can choose an element an+1 that is in I but that is not in a1R +...+ anR. In this way we can create an ascending chain of right ideals that is never stationary

a1R < a1R+a2R < ...

2 ==> 3. Let S be a nonempty set of right ideals that does not have a maximal element. Then given a right ideal I in S there must exist another J in S such that I is contained in but does not equal J. Repeating this we obtain an ascending chain of right ideals that is never stationary.

3 ==> 2. The maximum condition applied to the set of right ideals in the chain shows that the chain is eventually stationary.

A cookie to anyone who can tell me where we sneakily used the axiom of choice in the above proof. (Actually only the axiom of dependent choice is needed.)

Examples of Noetherian rings

  • Any field is Noetherian since any ideal must either be {0} or R. Similarly any division ring is Noetherian. So this applies to Q, R, C, H, for example.
  • Any principal ideal domain is Noetherian (since this is a commutative ring in which every ideal needs just one generator). This applies to Z and k[x], for a field k.
  • More generally the celebrated The Hilbert Basis Theorem tells us that the polynomial ring in several variables over any commutative Noetherian ring is Noetherian.

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