Let $A$ be an abelian variety defined over a number field $K$. Let $\mathfrak{p}$ be a prime of $K$ for which $A$ has good reduction and let $k=\mathcal{O}_{K,\mathfrak{p}}/\mathfrak{p}$. Let $\mathcal{A}$ be a Neron model for $A$ over $\mathcal{O}_{K,\mathfrak{p}}$. Then I have seen it mentioned that the following composition is an injection:
$$\operatorname{End}_K(A)= \operatorname{End}_{\mathcal{O}_{K,\mathfrak{p}}} (A)\to \operatorname{End}_k(\mathcal{A}\otimes k).$$
I cannot however seem to prove this. It seems like it should be easy though. Any help or direction is appreciated.
One may assume that $A$ is simple over $K$. Then $\mathrm{End}_K(A)$ is a division ring. Since the morphism into $\mathrm{End}_k(A\otimes k)$ is not zero, the kernel is $0$.