A classification of the minimal ring extensions of an integral domain

Let R be any integral domain. The minimal (commutative unital) ring extensions S of R are, up to R-algebra isomorphism, of three nonoverlapping types: (i) the domains S that contain R and are minimal ring extensions of R; (ii) the idealizations R(+)R/M arising from maximal ideals M of R; and (iii) the direct products R×R/M arising from maximal ideals M of R. Distinct maximal ideals of R lead to nonisomorphic idealizations (respectively direct products) in case (ii) (respectively case (iii)). If R is not a field, then distinct domains S arising in case (i) within the same quotient field of R are not isomorphic as R-algebras. If R is a field K, then the domains S arising in (i) are the minimal (necessarily algebraic) field extensions of K; in this case, distinct such fields S1,S2 are K-algebra isomorphic if and only if Si=K(αi) where α1,α2 are roots of the same irreducible polynomial in K[X].