Definition:Vector Space over Subring

Theorem
Let $$K$$ be a division subring of the division ring $$\left({L, +_L, \times_L}\right)$$.

Let $$\left({G, +_G: \circ}\right)_L$$ be a $L$-vector space.

Then $$\left({G, +_G: \circ_K}\right)_K$$ is a $K$-vector space, where $$\circ_K$$ is the restriction of $$\circ$$ to $$K \times G$$.

The module $$\left({G, +_G: \circ_K}\right)_K$$ is called the $$K$$-module obtained from $$\left({L, +_L, \times_L}\right)$$ by restricting scalar multiplication.

Compare Subring Module.