field theory - In what vector space does Dedekind's lemma live?

Dedekind's lemma in field theory says this:

Let $E$ and $L$ be fields, and $\sigma_1,\ldots,\sigma_n:E\longrightarrow L$ be distinct field homomorphisms. Then $\sigma_1,\ldots,\sigma_n$ are $L$-linearly independent, that is $$\sum_{i=1}^na_i\sigma_i=0,\;a_i\in L\implies (\forall i)\;a_i=0,$$

Is there an $L$-vector space in which this linear independence takes place? All field homomorphisms from $E$ to $L$ don't constitute a vector space because there is no neutral element in this set. Also, the sum of two homomorphisms may not be a homomorphism. If $\operatorname{char}L=2,$ and $\sigma:E\longrightarrow L$ is a field homomorphism, then $\sigma+\sigma=0$ isn't a field homomorphism. I'm not really sure when the sum of two homomorphisms is again a homomorphism. Also, I'm not sure when the additive inverse of a homomorphism is again a homomorphism. Could you please help with these questions?

Answer

The space of all additive homomorphisms from $E$ to $L$ ($E$ and $L$ are considered as abelian groups w.r.t. addition). If $f$ is such homomorphism and $a\in L$, then $af$ maps $x$ to $a\cdot f(x)$.