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
- Zero-dimensional ring
- One-dimensional ring
- Local Noetherian ring
- Multivariate polynomial ring over a field
Conjunction with other properties
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.