Extending scalars from $k$ to $K$: how find $K$ linear maps?

Question

If I extend scalars from a ring $R$ to a ring $S$ by a homomorphism $f:R \to S$, then starting with an $R$ module $M$, I get an $S$ module $S \otimes_R M$. Given $\sigma \in \text{End}_R(M)$, I know that $1 \otimes \sigma \in \text{End}_S(S \otimes _R M)$. But is there any other way to build elements of $\text{End}_S(S \otimes_R N)$?

I ask because I'm thinking about extending scalars from $k$ to $K$ on a vector space, and I'm trying to get a hold on $\text{End}_K(V \otimes_k K)$, besides just elements of the form $\sigma \otimes 1$, for $\sigma \in \text{End}_k(V)$.

Answer 1

We have $$\text{End}_S(S \otimes _R M)=\text{Hom}_S(S \otimes _R M,S \otimes _R M)\simeq\text{Hom}_R(M,S \otimes _R M)\simeq\text{Hom}_R(M,M)\otimes _R S=\text{End}_R(M)\otimes_RS$$ when $S$ is $R$-flat and $M$ finitely presented; see Hom and tensor with a flat module.

In general it's not true, but for the case of finitely dimensional vector spaces we can't have other endomorphisms.