Noetherian ring: Difference between revisions

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In [[algebra]], a '''Noetherian ring''' is a [[ring (mathematics)|ring]] with a condition on the [[lattice (order)|lattice]] of [[ideal]]s.
In [[algebra]], a '''Noetherian ring''' is a [[ring (mathematics)|ring]] with a condition on the [[lattice (order)|lattice]] of [[ideal]]s.


==Definition==
==Definition==
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When the above conditions are satisfied, <math>A</math> is said to be ''Noetherian''.  Alternatively, the ring <math>A</math> is Noetherian if is a [[Noetherian module]] when regarded as a module over itself.
When the above conditions are satisfied, <math>A</math> is said to be ''Noetherian''.  Alternatively, the ring <math>A</math> is Noetherian if is a [[Noetherian module]] when regarded as a module over itself.
A '''Noetherian domain''' is a Noetherian ring which is also an [[integral domain]].


==Examples==
==Examples==
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#The [[localization]] of <math>A</math> by a multiplicative subset <math>S</math> is again Noetherian.  
#The [[localization]] of <math>A</math> by a multiplicative subset <math>S</math> is again Noetherian.  
#'''Hilbert's Basis Theorem''': The [[polynomial ring]] <math>A[X]</math> is Noetherian (hence so is <math>A[X_1,\ldots,X_n]</math>).
#'''Hilbert's Basis Theorem''': The [[polynomial ring]] <math>A[X]</math> is Noetherian (hence so is <math>A[X_1,\ldots,X_n]</math>).
==See also==
{{r|Artinian ring}}
{{r|Noetherian module}}


==References==
==References==
* {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 | pages=186-187 }}
* {{cite book | author=Serge Lang | authorlink=Serge Lang | title=Algebra | edition=3rd ed | publisher=[[Addison-Wesley]] | year=1993 | isbn=0-201-55540-9 | pages=186-187 }}[[Category:Suggestion Bot Tag]]

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In algebra, a Noetherian ring is a ring with a condition on the lattice of ideals.

Definition

Let be a ring. The following conditions are equivalent:

  1. The ring satisfies an ascending chain condition on the set of its ideals: that is, there is no infinite ascending chain of ideals .
  2. Every ideal of is finitely generated.
  3. Every nonempty set of ideals of has a maximal element when considered as a partially ordered set with respect to inclusion.

When the above conditions are satisfied, is said to be Noetherian. Alternatively, the ring is Noetherian if is a Noetherian module when regarded as a module over itself.

A Noetherian domain is a Noetherian ring which is also an integral domain.

Examples

Useful Criteria

If is a Noetherian ring, then we have the following useful results:

  1. The quotient is Noetherian for any ideal .
  2. The localization of by a multiplicative subset is again Noetherian.
  3. Hilbert's Basis Theorem: The polynomial ring is Noetherian (hence so is ).

References