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Noetherian ring

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In mathematics, a ring is called Noetherian if, intuitively speaking, its ideals are not "too large", expressed by a certain finiteness condition.

Formally, the ring R is left-Noetherian iff one (and therefore all) of the following equivalent conditions hold:

  • Every left ideal in R is finitely generated.
  • Any ascending chain I1I2I3 ⊆ ... of left ideals in R eventually becomes stationary: there exists a natural number n such that Im = In for all mn. This can be rephrased as "the poset of (two-sided) ideals in R under inclusion has the ascending chain condition".
  • Any non-empty set of left ideals of R has a maximal element.
The ring R is called right-Noetherian if the above conditions are true for right ideals, and it is called Noetherian if it is both left-Noetherian and right-Noetherian. For commutative rings, these three notions coincide.

Every field is trivially Noetherian, since a field F has only two ideals - F and {0}. Every finite ring is Noetherian. Other familar examples of Noetherian rings are the ring of integers, Z; and Z[x], the ring of polynomials over the integers. In fact, the Hilbert basis theorem states that if a ring R is Noetherian, then the polynomial ring R[x] is Noetherian as well. If R is a Noetherian ring and I is an ideal, then the quotient ring R/I is also Noetherian. Every commutative Artinian ring[?] is Noetherian.

An example of a ring that's not Noetherian is a ring of polynomials in infinitely many variables: the ideal generated by these variables cannot be finitely generated.

Noetherian rings are named after the mathematician Emmy Noether, who developed much of their theory.



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