# Difference between revisions of "Noetherian ring"

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## Revision as of 02:28, 18 July 2013

This article is about a standard (though not very rudimentary) definition in commutative algebra. The article text may, however, contain more than just the basic definition

View a complete list of semi-basic definitions on this wiki

This article defines a property of commutative unital rings; a property that can be evaluated for a commutative unital ring

View all properties of commutative unital ringsVIEW RELATED: Commutative unital ring property implications | Commutative unital ring property non-implications |Commutative unital ring metaproperty satisfactions | Commutative unital ring metaproperty dissatisfactions | Commutative unital ring property satisfactions | Commutative unital ring property dissatisfactions

## Definition

### Symbol-free definition

A commutative unital ring is termed Noetherian if it satisfies the following equivalent conditions:

- Ascending chain condition on ideals: Any ascending chain of ideals stabilizes after a finite length.
- Every ideal in the ring is finitely generated.
- Every prime ideal in the ring is finitely generated.

### Definition with symbols

A commutative unital ring is termed a **Noetherian ring** if it satisfies the following equivalent conditions:

- If is a well-ordered set and are ideals such that for , there exists some such that for all .
- If is an ascending chain of ideals, then there exists such that for all .
- For every ideal in , there exist elements such that is the ideal generated by .
- For every prime ideal in , there exist elements such that equals the ideal generated by .

## Relation with other properties

### Conjunction with other properties

Conjunction | Other component of conjunction | Additional comments |
---|---|---|

Noetherian domain | integral domain | |

reduced Noetherian ring | reduced ring: it has no nonzero nilpotent elements. | |

Noetherian normal domain | normal domain | |

Noetherian unique factorization domain | unique factorization domain | |

local Noetherian ring | local ring | |

local Noetherian domain | local domain | |

zero-dimensional Noetherian ring | zero-dimensional ring: every prime ideal in it is a maximal ideal | |

one-dimensional Noetherian domain | one-dimensional domain: integral domain in which every nonzero prime ideal is maximal | |

finite-dimensional Noetherian ring | finite-dimensional ring: its Krull dimension is finite. |

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Polynomial ring over a field | for a field | click here | ||

Artinian ring | descending chain of ideals stabilizes eventually | Artinian implies Noetherian | Noetherian not implies Artinian | click here |

Principal ideal ring | every ideal is principal | principal ideal ring implies Noetherian | Noetherian not implies principal ideal ring | click here |

Dedekind domain | click here | |||

Cohen-Macaulay ring | click here | |||

Affine ring | click here |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Coherent ring | click here |

## Metaproperties

### Closure under taking the polynomial ring

This property of commutative unital rings is polynomial-closed: it is closed under the operation of taking the polynomial ring. In other words, if is a commutative unital ring satisfying the property, so isView other polynomial-closed properties of commutative unital rings

The polynomial ring over a Noetherian ring is again Noetherian. This is a general formulation of the Hilbert basis theorem, which asserts in particular that the polynomial ring over a field is Noetherian.
`Further information: Noetherianness is polynomial-closed`

### Closure under taking quotient rings

This property of commutative unital rings is quotient-closed: the quotient ring of any ring with this property, by any ideal in it, also has this propertyView other quotient-closed properties of commutative unital rings

The quotient ring of a Noetherian ring by an ideal, is also Noetherian. `Further information: Noetherianness is quotient-closed`

### Closure under taking subrings

This property of commutative unital rings isnotclosed under taking subrings; in other words, a subring of a commutative unital ring with this property need not have this property

A subring of a Noetherian ring is not necessarily Noetherian. For this, consider any non-Noetherian integral domain; this is a subring of a field, which is Noetherian.

### Closure under taking localizations

This property of commutative unital rings is closed under taking localizations: the localization at a multiplicatively closed subset of a commutative unital ring with this property, also has this property. In particular, the localization at a prime ideal, and the localization at a maximal ideal, have the property.

View other localization-closed properties of commutative unital rings

A localization of a Noetherian ring is Noetherian. Intuitively, when we take localizations, we land up with fewer ideals, so the ascending chain condition becomes easier to satisfy. `Further information: Noetherianness is localization-closed`

### Direct products

This property of commutative unital rings is finite direct product-closed: a finite direct product of rings with this property, also has this property

View other finite direct product-closed properties of commutative unital rings

A direct product of finitely many Noetherian rings is again Noetherian. This is essentially because ideals in the direct product look like direct products of ideals in the factors. *For full proof, refer: Noetherianness is finite direct product-closed*

### Closure under taking completions

This property of commutative unital rings is completion-closed: the completion of a ring with this property, at any maximal ideal, also has this property

View other completion-closed properties of commutative unital rings

The completion of a Noetherian ring at a maximal ideal is again a Noetherian ring. *For full proof, refer: Noetherianness is completion-closed*

- Semi-basic definitions in commutative algebra
- Standard terminology
- Properties of commutative unital rings
- Polynomial-closed properties of commutative unital rings
- Quotient-closed properties of commutative unital rings
- Localization-closed properties of commutative unital rings
- Finite direct product-closed properties of commutative unital rings
- Completion-closed properties of commutative unital rings