Let $R$ be a commutative unital ring that is a DVR. Let $\mathfrak{m}$ be the unique maximal ideal of $R$. Assuming that $\mathrm{char}(R)=\mathrm{char}(R/\mathfrak{m})$, does there necessarily exist a homomorphism $R/\mathfrak{m}\rightarrow R$ whose composition with the quotient map $R\rightarrow R/\mathfrak{m}$ is the identity?
I believe that this is true for $R\approx k[[x]]$, $k$ a field.
This is true for complete local rings by Cohen’s structure theorem (specifically, see [Matsumura, Theorem 28.3]). On the other hand, the following example shows that it is not true for local rings that are not complete, even for DVR’s:
Example [Matsumura, Exercise 28.2]. Let $k$ be an imperfect field of characteristic $p > 0$, and let $a \in k \smallsetminus k^p$. Consider the DVR $$R = k[X]_{(X^p-a)}.$$ Then, $R$ is a DVR with residue field $k(a^{1/p})$. On the other hand, $R$ does not have a subfield containing $k(a^{1/p})$, since $$k[X] \subseteq R \subseteq k(X),$$ and $k(X)$ does not contain a $p$th root of $a$.
