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
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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
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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.
References
[1] | B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German) |
[2] | S. Lang, "Algebra" , Addison-Wesley (1974) |
[3] | C. Faith, "Algebra: rings, modules, and categories" , 1 , Springer (1973) |
Noetherian ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Noetherian_ring&oldid=49330