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

Let $(R,M,k)$ be a complete local ring (not necessarily Noetherian). As the first main result of this article, we prove that in the unequal characteristic case $\text{Char}(R)\mathrm{\ne }\text{Char}(k)$, the natural surjective map between the groups of units $R^{\ast }\to k^{\ast }$ admits a splitting. Next, we reprove by a new method that $R$ is equi-characteristic, i.e., $\text{Char}(R)=\text{Char}(k)$ if and only if the natural surjective ring map $R\to k$ admits a splitting, or equivalently, $R$ has a coefficient field. In our proof there is no need for the existence of the coefficient fields for equi-characteristic complete local rings, whose existence is the most difficult part of the known proof. This is one of the main contributions of the article. \ As an application, we show that for any complete local ring $(R,M,k)$ 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. In particular, we have an isomorphism of Abelian groups $R^{\ast }\simeq (1+M)\times k^{\ast }$. We also show with an example that the above exact sequence does not split for many incomplete local rings.