# Noetherian ring

left (right)

A ring satisfying one of the following equivalent conditions:

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

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

3) every strictly ascending chain of left (or right) ideals in 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 be the ring of matrices of the form where is a rational integer and and are rational numbers, with the usual addition and multiplication. Then is right, but not left, Noetherian, since the left ideal of elements of the form 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 is a left Noetherian ring, then so is the polynomial ring . The corresponding property holds for the ring of formal power series over a Noetherian ring. In particular, polynomial rings of the form or , where is a field and the ring of integers, and also quotient rings of them, are Noetherian. Every Artinian ring is Noetherian. The localization of a commutative Noetherian ring relative to some multiplicative system is again Noetherian. If in a commutative Noetherian ring , is an ideal such that no element of the form , where , is a divisor of zero, then . This means that any such ideal defines on a separable -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.