nLab prevector space

Idea

In the same vein that commutative rings are to integral domains and GCD rings are to GCD domains, prevector spaces are to vector spaces.

Definition

A $R$-module $V$ is a prevector space if $R$ is a prefield ring. The elements of $V$ are called vectors.

Examples

• Every $\mathbb{Q}$-vector space is a $\mathbb{Q}$-prevector space.

• Every classical vector space is a $F$-prevector space for a classical field $F$ (defined using denial inequality).

• Every Heyting vector space is a $F$-prevector space for a Heyting field $F$.

• Every discrete vector space is a $F$-prevector space for a discrete field $F$.

• Every residue vector space is a $F$-prevector space for a residue field $F$.

See also

• module

• vector space

• super prevector space

• N-graded prevector space

• prefield ring