  Given two rings $R \subseteq S$, $S$ is said to be a minimal ring extension
of $R$ if $R$ is a maximal subring of $S$. In this article, we study minimal
extensions of an arbitrary ring $R$, with particular focus on those possessing
nonzero ideals that intersect $R$ trivially. We will also classify the minimal
ring extensions of prime rings, generalizing results of Dobbs, Dobbs & Shapiro,
and Ferrand & Olivier on commutative minimal extensions.
