Let $K$ be a field, $A$ be a commutative unital $K$-algebra and $V$ be a $K$-vector space. Is the $K$-linear map $$x\mapsto 1_A\otimes x:V\to A\otimes V$$ injective?
The specific case that I am interested in is when $K=\mathbb{F}_2$ and $a^2\in\lbrace 0,1\rbrace$ for all $a\in A$. (Could also restrict to $A$ being finite-dimensional.)
I'm sure there is something I'm missing, as I have no idea how to proceed in the general case or the more specific case.
Since $A$ is a $K$-algebra, it is equipped with a ring map $K \xrightarrow{f} A : x \mapsto x.1_A$. In particular, $f$ is $K$-linear. Further, $K$ is a field implies that $f$ is injective.
Now, $V$ being a $K$-vector space implies that it is a flat $K$-module (since vector spaces are free modules and hence flat).
Thus, the functor $\_ \otimes_K V : K-Mod \rightarrow K-Mod$ preserves monics. Hence, the image of the arrow $f$ under this functor, namely $f \otimes 1_V : V \cong K \otimes_K V \rightarrow A \otimes_K V$ is a monic in the category $K$-Mod, that is an injection. And this map is precisely $x \mapsto 1_A \otimes x$ from $V$ into $A \otimes_K V$.