# Finite-dimensional algebra over a field

From Commalg

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

*Any integral domain satisfying this property of commutative unital rings, must be a field*

## Definition

A **finite-dimensional algebra over a field** is a commutative unital ring that contains a subfield, such that the ring is finite-dimensional, when viewed as a vector space over the field. The dimension here is not to be confused with the Krull dimension, which is always zero for such algebras.