# Noetherian ring

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A Noetherian ring is a ring all of whose ideals are finitely generated.

## Formal definition

A ring R is said to be a Noetherian ring if every ideal of R is finitely generated.

## Examples

• The rings $\mathbb{Z}$ , $\mathbb{Q}$ , $\mathbb{R}$ , and $\mathbb{C}$ are all examples of Noetherian rings.
• Any field is a Noetherian ring, because its only ideals are the zero ideal and the field itself, which are both obviously finitely generated. In fact, a field is an Artinian ring.
• Any finite ring is a Noetherian ring. In fact, a finite ring is an Artinian ring. In particular, for any positive integer n, the integers mod n form a Noetherian ring (in fact, an Artinian ring).
• The ring of polynomials in any number of variables with real coefficients is a Noetherian ring.

## Non-examples

• The ring of functions from the real numbers to itself is not a Noetherian ring.
• The ring of all algebraic integers is another example of a non-Noetherian ring.
• The ring of polynomials in one variable x with rational coefficients whose constant term is an integer is a third example of a non-Noetherian ring.