Vector Space over Subring

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


Proof


Also see


Sources