Let $R$ be a complete, Noetherian, local ring with maximal ideal $\mathfrak{m}$ and residue field $R/\mathfrak{m} \cong \mathbb{F}_p$. Note that by "complete", we mean that $$ R \cong \varprojlim_n R/\mathfrak{m}^n. $$
I am trying to show that $R$ is an algebra over the $p$-adic integers $\mathbb{Z}_p$.
My attempt
Since we want to construct a ring homomorphism $\mathbb{Z}_p \to R$, I would like to construct compatible homomorphisms $$ R \to \mathbb{Z}/p^n\mathbb{Z}. $$ which would give a cone on the inverse system used to define $$ \mathbb{Z}_p = \varprojlim_n \mathbb{Z}/p^n\mathbb{Z}, $$ and hence induce a homomorphism $\mathbb{Z}_p \to R$ be the universal property of limits. For $n=1$, there is an obvious candidate, namely the quotient map $\varphi_1:R \to R/\mathfrak{m} \cong \mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}$.
Now we would like a homomorphism $\varphi_2:R \to \mathbb{Z}/p^2\mathbb{Z}$ such that the composition $$ R \overset{\varphi_2}{\to} \mathbb{Z}/p^2\mathbb{Z} \to \mathbb{Z}/p\mathbb{Z} $$ is the same as $\varphi_1$. Now, I would like it to be the case that $R/\mathfrak{m}^n =\cong \mathbb{Z}/p^n\mathbb{Z}$, which would allows us to define all higher maps the same way we definded $\varphi_1$. However, I don't see why this would be the case, and I'm not sure how to proceed.
Any help would be much appreciated.
As @DanielHast hints in his comment, you’re looking for a nonexistent thing. There’s no ring homomorphism from $R=\Bbb Z_p[\sqrt p\,]$ to $\Bbb Z_p$, for instance, even though this $R$ definitely is the kind of complete ring you’re considering. The maximal ideal of $R$ is $\mathfrak m=\langle\sqrt p, p\rangle$, where the angle brackets denote an enumeration of the $\Bbb Z_p$-basis. And it is not true that $R/\mathfrak m^2\cong \Bbb Z_p/(p)^2$.
What you want is a map $\Bbb Z_p\to R\,$; that’s what makes $R$ a $\Bbb Z_p$-algebra. You can construct it by taking the necessary map $\Bbb Z\to R$, and then showing that $p$-adically convergent sequences of elements of $\Bbb Z$ get carried to $\mathfrak m$-adically convergent sequences of elements of $R$. As I recall, it may be rather tiresome to verify that you’ve defined a good ring morphism $\Bbb Z_p\to R$.