# Noetherian ring

left (right)

A ring $A$ satisfying one of the following equivalent conditions:

1) $A$ is a left (or right) Noetherian module over itself;

2) every left (or right) ideal in $A$ has a finite generating set;

3) every strictly ascending chain of left (or right) ideals in $A$ breaks off after finitely many terms.

An example of a Noetherian ring is any principal ideal ring, i.e. a ring in which every ideal has one generator.

Noetherian rings are named after E. Noether, who made a systematic study of such rings and carried over to them a number of results known earlier only under more stringent restrictions (for example, Lasker's theory of primary decompositions).

A right Noetherian ring need not be left Noetherian and vice versa. For example, let $A$ be the ring of matrices of the form

$$\left \| \begin{array}{cc} a &\alpha \\ 0 &\beta \\ \end{array} \right \| ,$$

where $a$ is a rational integer and $\alpha$ and $\beta$ are rational numbers, with the usual addition and multiplication. Then $A$ is right, but not left, Noetherian, since the left ideal of elements of the form

$$\left \| \begin{array}{cc} 0 &\alpha \\ 0 & 0 \\ \end{array} \right \|$$

does not have a finite generating set.

Quotient rings and finite direct sums of Noetherian rings are again Noetherian, but a subring of a Noetherian ring need not be Noetherian. For example, a polynomial ring in infinitely many variables over a field is not Noetherian, although it is contained in its field of fractions, which is Noetherian.

If $A$ is a left Noetherian ring, then so is the polynomial ring $A [ X ]$. The corresponding property holds for the ring of formal power series over a Noetherian ring. In particular, polynomial rings of the form $K [ X _ {1} \dots X _ {n} ]$ or $\mathbf Z [ X _ {1} \dots X _ {n} ]$, where $K$ is a field and $\mathbf Z$ the ring of integers, and also quotient rings of them, are Noetherian. Every Artinian ring is Noetherian. The localization of a commutative Noetherian ring $A$ relative to some multiplicative system $S$ is again Noetherian. If in a commutative Noetherian ring $A$, $\mathfrak m$ is an ideal such that no element of the form $1 + m$, where $m \in \mathfrak m$, is a divisor of zero, then $\cap _ {k=} 1 ^ \infty \mathfrak m ^ {k} = 0$. This means that any such ideal $\mathfrak m$ defines on $A$ a separable $\mathfrak m$- adic topology. In a commutative Noetherian ring every ideal has a representation as an incontractible intersection of finitely many primary ideals. Although such a representation is not unique, the number of ideals and the set of prime ideals associated with the given primary ideals are uniquely determined.

How to Cite This Entry:
Noetherian ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noetherian_ring&oldid=49490
This article was adapted from an original article by L.V. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article