Splittings in a complete local ring and decomposition its group of units

Let $(R,M,k)$ be a complete local ring (not necessarily Noetherian). Then we prove that the natural surjective ring map $R\rightarrow k$ admits a splitting if and only if $\Char(R)=\Char(k)$. In addition, if $\Char(R)\neq\Char(k)$, then we prove that the natural surjective map between the groups of units $R^{\ast}\rightarrow k^{\ast}$ admits a splitting. As an application of this theorem, we show that the following short exact sequence of Abelian groups: $$\xymatrix{1\ar[r]&1+M\ar[r]& R^{\ast}\ar[r]&k^{\ast} \ar[r]&1}$$ is always split. We also show with an example that the above exact sequence does not split for most incomplete local rings. In an attempt to prove the above main theorem, we first obtained a characterization for separability which states that an algebraic extension of fields is separable if and only if it is formally étale. In particular, in the characteristic zero, every algebraic extension of fields is formally étale. Next, we show that every field extension over a perfect field is formally smooth.