To make use of the Lie algebra action of $\mathsf{gl}_2(\mathbb{C})$ to establish a isomorphism in modular representation theory, I would like an answer to this question:

Let $K$ be a field of prime characteristic. When is there a subring $R$ of the complex numbers and a maximal ideal $M$ of $R$ such that $R/M \cong K$?

Clearly no such ring $R$ exists if $K$ has strictly more than $|\mathbb{C}|$ independent transcendental elements. Is this the only obstruction? Is there a reasonably explicit way to construct a suitable $R$ when $K$ is the algebraic closure of $\mathbb{F}_p$?

As a follow-up (which at first I thought I needed, but I now see I can get around by working with $\mathrm{GL}_2(\mathbb{C})$ rather than $\mathrm{SL}_2(\mathbb{C})$), note that if $R/M \cong K$ then the induced map $\mathrm{GL}_d(R) \rightarrow \mathrm{GL}_d(K)$, defined on a $d \times d$ matrix with entries in $R$ by applying $R \twoheadrightarrow R/M \cong K$ to each entry, is a surjective group homomorphism.

Is is true in general that the restriction of the group homomorphism $\mathrm{GL}_d(R) \rightarrow \mathrm{GL}_d(K)$ to $\mathrm{SL}_d(R)$ is surjective onto $\mathrm{SL}_d(K)$, or are there further obstructions?

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