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>). | ||
==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]] |
Latest revision as of 12:00, 26 September 2024
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:
- The ring satisfies an ascending chain condition on the set of its ideals: that is, there is no infinite ascending chain of ideals .
- Every ideal of is finitely generated.
- 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
- A field is Noetherian, since its only ideals are (0) and (1).
- A principal ideal domain is Noetherian, since every ideal is generated by a single element.
- The ring of integers Z
- The polynomial ring over a field
- The ring of continuous functions from R to R is not Noetherian. There is an ascending sequence of ideals
Useful Criteria
If is a Noetherian ring, then we have the following useful results:
- The quotient is Noetherian for any ideal .
- The localization of by a multiplicative subset is again Noetherian.
- Hilbert's Basis Theorem: The polynomial ring is Noetherian (hence so is ).
References
- Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 186-187. ISBN 0-201-55540-9.