I don't know much about the proof of the Freyd–Mitchell embedding theorem and I could not find an answer to my question looking naïvely online, but at the same time I feel like this is the kind of question to which someone who knows some of the details of the proof might be able to answer immediately, so it's probably worth trying. Here it is:

Can the Freyd-Mitchell embedding theorem be made stronger for $k$-linear abelian categories (where $k$ is a field), saying that not only, if $\mathcal{A}$ is a small abelian $k$-linear category, there exists a ring $R$ and a full, faithful, exact functor $F: \mathcal{A} → \text{$R$-$\mathrm{Mod}$}$, but that, moreover, $R$ can be assumed to be a $k$-algebra and $F$ to be $k$-linear?

 

More in general (also for non-$k$-linear categories): can one say anything about $R$? Is there even a unique "minimal" $R$ (up to Morita equivalence)?

I don't know much about the proof of the Freyd–Mitchell embedding theorem and I could not find an answer to my question looking naïvely online, but at the same time I feel like this is the kind of question to which someone who knows some of the details of the proof might be able to answer immediately, so it's probably worth trying. Here it is:

Can the Freyd-Mitchell embedding theorem be made stronger for $k$-linear abelian categories (where $k$ is a field), saying that not only, if $\mathcal{A}$ is a small abelian $k$-linear category, there exists a ring $R$ and a full, faithful, exact functor $F: \mathcal{A} → \text{$R$-$\mathrm{Mod}$}$, but that, moreover, $R$ can be assumed to be a $k$-algebra and $F$ to be $k$-linear?

More in general (also for non-$k$-linear categories): can one say anything about $R$? Is there even a unique "minimal" $R$ (up to Morita equivalence)?

I don't know much about the proof of the Freyd-Mitchell embedding theorem and I could not find an answer to my question looking naively online, but at the same time I feel like this is the kind of question to which someone who knows some of the details of the proof might be able to answer immediately, so it's probably worth trying. Here it is:

Can the Freyd-Mitchell embedding theorem be made stronger for $k-$linear abelian categories (where $k$ is a field), saying that not only, if $\mathcal{A}$ is a small abelian $k-$linear category, there exists a ring $R$ and a full, faithful, exact functor $F: \mathcal{A} → R-\mathrm{Mod}$, but that, moreover, $R$ can be assumed to be a $k-$algebra and $F$ to be $k-$linear?

More in general (also for non$-k-$linear categories): can one say anything about $R$? Is there even a unique "minimal" $R$ (up to Morita equivalence)?

I don't know much about the proof of the Freyd–Mitchell embedding theorem and I could not find an answer to my question looking naïvely online, but at the same time I feel like this is the kind of question to which someone who knows some of the details of the proof might be able to answer immediately, so it's probably worth trying. Here it is:

Can the Freyd-Mitchell embedding theorem be made stronger for $k$-linear abelian categories (where $k$ is a field), saying that not only, if $\mathcal{A}$ is a small abelian $k$-linear category, there exists a ring $R$ and a full, faithful, exact functor $F: \mathcal{A} → \text{$R$-$\mathrm{Mod}$}$, but that, moreover, $R$ can be assumed to be a $k$-algebra and $F$ to be $k$-linear?

More in general (also for non-$k$-linear categories): can one say anything about $R$? Is there even a unique "minimal" $R$ (up to Morita equivalence)?