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.

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)