Finite-dimensional ring

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 rings
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

This property of commutative unital rings is completely determined by the spectrum, viewed as an abstract topological space. The corresponding property of topological spaces is: finite bound on length of ascending chains of irreducible closed subsets

View other properties of commutative unital rings determined by the spectrum

Definition

Symbol-free definition

A finite-dimensional ring is a ring whose Krull dimension is finite.

Relation with other properties

Stronger properties

• Artinian ring
• Zero-dimensional ring
• One-dimensional ring
• Local Noetherian ring
• Multivariate polynomial ring over a field

Conjunction with other properties

• Finite-dimensional Noetherian ring

Metaproperties

Polynomial-closedness

In general, a polynomial ring over a finite-dimensional ring need not be finite-dimensional; when the ring is Noetherian, however, the polynomial ring is finite-dimensional, with dimension exactly one more than that of the original ring.