I came across the following definition:
Let $\Omega$ be algebraically closed, $k \subseteq \Omega$ a subfield, and $V$ an $\Omega$-vector space. A $k$-structure on $V$ is a $k$-vector space $V_k$ such that the natural map $$V_k \otimes_k \Omega \rightarrow V$$ produces an isomorphism of vector spaces. Then we will say $V$ is defined over $k$ by means of $V_k$.
The definition makes no sense, because there is no "natural map" without some $k$-vector space homomorphism $V_k \rightarrow V$. My guess would be then that a $k$-structure should be defined as a pair $(V_k, \phi)$, where $\phi$ is a $k$-homomorphism of $V_k$ into $V$. My question then, is should we always assume this homomorphism is injective? In other words, is $V_k$ always taken to be a $k$-submodule of $V$ (if we identify $V_k$ with its image under $\phi$)? In that case, $V_k$ would be a $k$-structure if and only if the $\Omega$-module generated by $V_k$ is $V$.
If $\Phi:V_k \otimes_k \Omega \to V$ is injective, then $\phi:V_k \to V$ is injective, because $\phi$ is just the restriction of $\Phi$ to tensors of the form $v\otimes 1$.
So we might as well take $V_k$ in the first place to be a $k$-subspace of $V$.