Suppose $R$ and $S$ are rings such that $R\subset S$ with unique maximal ideals $m_R$ and $m_S$, respectively. Suppose $m_R=m_S\cap R$. Is it true then that $S/m_S$ is a field extension of $R/m_R$?
This is just a generalization of a specific example of something in the exposition of a text I'm reading. It doesn't seem that we can just use the lattice isomorphism theorem since the quotients are different.