Definition 10.160.4. Let $(R, \mathfrak m)$ be a complete local ring. A subring $\Lambda \subset R$ is called a coefficient ring if the following conditions hold:
$\Lambda $ is a complete local ring with maximal ideal $\Lambda \cap \mathfrak m$,
the residue field of $\Lambda $ maps isomorphically to the residue field of $R$, and
$\Lambda \cap \mathfrak m = p\Lambda $, where $p$ is the characteristic of the residue field of $R$.