Division Ring is Vector Space over Prime Subfield

Theorem

Let $\struct {K, +, \times}$ be a division ring.

Let $\struct {S, +, \times}$ be the prime subfield of $K$

Then $\struct {K, +, \times_S}_S$ is an $S$-vector space, where $\times_S$ is the restriction of $\times$ to $S \times K$.

Proof

Because $K$ is a division ring, it satisfies the vector space axioms for addition, in particular it is an abelian group. The distribuitivty and associativity of multiplication follow from the rules for multiplciation in any ring. Also, any prime field contains the multiplicative identity.

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$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules: Example $26.2$