Finite-dimensional 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
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

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.

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 property


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