Noetherian Ring - Examples

Examples

• Any field, including fields of rational numbers, real numbers, and complex numbers. (A field only has two ideals — itself and (0).)
• Any principal ideal domain, such as the integers, is Noetherian since every ideal is generated by a single element.
• A Dedekind domain (e.g., rings of integers) is Noetherian since every ideal is generated by at most two elements.
• The coordinate ring of an affine variety is a noetherian ring, as a consequence of the Hilbert basis theorem.
• The enveloping algebra U of a finite-dimensional Lie algebra is a both left and right noetherian ring; this follows from the fact that the associated graded ring of U is a quotient of, which is a polynomial ring over a field; thus, noetherian.
• The ring of polynomials in finitely-many variables over the integers or a field.

Rings that are not Noetherian tend to be (in some sense) very large. Here are two examples of non-Noetherian rings:

• The ring of polynomials in infinitely-many variables, X₁, X₂, X₃, etc. The sequence of ideals (X₁), (X₁, X₂), (X₁, X₂, X₃), etc. is ascending, and does not terminate.
• The ring of continuous functions from the real numbers to the real numbers is not Noetherian: Let I_n be the ideal of all continuous functions f such that f(x) = 0 for all x ≥ n. The sequence of ideals I₀, I₁, I₂, etc., is an ascending chain that does not terminate.

However, a non-Noetherian ring can be a subring of a Noetherian ring:

• The ring of rational functions generated by x and y/xn over a field k is a subring of the field k(x,y) in only two variables.

Indeed, there are rings that are left Noetherian, but not right Noetherian, so that one must be careful in measuring the "size" of a ring this way.

A unique factorization domain is not necessarily a noetherian ring. It does satisfy a weaker condition: the ascending chain condition on principal ideals.