Is there a name for this "homomorphism" between vector spaces over different fields?

Question

Let $V,W$ be vector spaces over fields $F_V,F_W$ respectively. If $F_V=F_W=F$ we have that a function $h:V\to W$ is a homomorphism from $V$ to $W$ if $h(v+w)=h(v)+h(w)$ and $h(\alpha v)=\alpha h(v)$, for all $v,w\in V$ and $\alpha\in F$.

But what if $F_V\neq F_W$? Then a "homomorphism" from $V$ to $W$ should be defined as two functions $h:V\to W$ and $g:F_V\to F_W$ such that $h$ and $g$ are homomorphisms and the following holds for all $v,w\in V$ and $\alpha\in F_V$:

$$h(v+w)=h(v)+h(w)$$

and

$$h(\alpha v)=g(\alpha)h(v)$$

Is this a valid "homomorphism" from $V$ to $W$, or two vector spaces over different fields cannot have the same structure?

Any field homomorphism is either zero or an embedding. So really you're talking about a field extension along with a vector space homomorphism. — Matthew Leingang, Feb 26 '19 at 16:31
@MatthewLeingang Any field homomorphism is an embedding. The constant $0$ map does not preserve $1$ (since $1\neq 0$ in a field). — Alex Kruckman, Feb 26 '19 at 16:38
It seems like $h=f\circ k$ where $k$ is an idempotent automorphism on $F_V$ and $f$ is a 1 to 1 homomorphism from the range of $k$ to a subspace of $F_W$. — irchans, Feb 26 '19 at 16:38
@AlexKruckman You're right, I was thinking of a homomorphism of irreducible modules. — Matthew Leingang, Feb 26 '19 at 17:12

score 2 · Accepted Answer · answered Feb 26 '19 at 16:51

Similar questions have been asked several times before:

You'll find good references in all of these links. In particular, the notion of homomorphism you write down is (1) the natural notion of homomorphism between models of the two-sorted theory of vector spaces over an arbitrary field (in the sense of model theory), (2) closely related to the notion of a semi-linear map between vector spaces, (3) an example of a fibered category construction.

Let me also point out that any such morphism can be factored into two stages. First, pick a field extension $g\colon F_V\to F_W$ (as Matthew Leingang points out in the comments, every homomorphism of fields is injective). This gives $F_W$ the structure of an $F_V$-algebra, and we can form the tensor product $V' = (F_W\otimes_{F_V} V)$. This is the "extension of scalars" which turns $V$ into an $F_W$-vector space. Finally, pick an $F_W$-linear map $h'\colon V'\to W$.

It turns out that the pairs $(g,h')$, where $g\colon F_V\to F_W$ is a field extension and $h'\colon V'\to W$ is an $F_W$-linear map, are in natural bijection with the morphisms $(g,h)$ in the sense of your question.

Great answer with relevant links and the final result included. Thanks for citing me even though I got it only half right! — Matthew Leingang, Feb 26 '19 at 17:14

Is there a name for this "homomorphism" between vector spaces over different fields?

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