Finite-dimensional Noetherian ring

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

A finite-dimensional Noetherian ring is a commutative unital ring that is both finite-dimensional (i.e. its Krull dimension is finite) and Noetherian (i.e. it satisfies the ascending chain condition on all ideals).

Relation with other properties

Stronger properties

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 R is a commutative unital ring satisfying the property, so is R[x]


View other polynomial-closed properties of commutative unital rings

The polynomial ring over a finite-dimensional Noetherian ring has dimension one more than the original ring.