Definition 2

Let A be a separated and complete local ring, whose residue field is of characteristic p. A Cohen subring of A means a subring C of A which is a $p$-ring such that $A = m_A + C$ (i.e. $A/m_A = C/(m_A \cap C)$).

If C is a Cohen subring of A, the ideal $m_A \cap C$ of C is maximal, hence equal to $pC$. The canonical mapping of $\kappa_C = C/pC$ onto $\kappa_A = A/m_A$ is therefore a field isomorphism.