A classification of the minimal ring extensions of certain commutative rings

Abstract

All rings considered are commutative with identity and all ring extensions are unital. Let R be a ring with total quotient ring T. The integral minimal ring extensions of R are catalogued via generator-and-relations. If T is von Neumann regular and no maximal ideal of R is a minimal prime ideal of R, the minimal ring extensions of R are classified, up to R-algebra isomorphism, as the minimal overrings (within T) of R and, for maximal ideals M of R, the idealizations R ( + ) R / M and the direct products R × R / M . If T is von Neumann regular, the minimal ring extensions of R in which R is integrally closed are characterized as certain overrings, up to R-algebra isomorphism, in terms of Kaplansky transforms and divided prime ideals, generalizing work of Ayache on integrally closed domains; no restriction on T is needed if R is quasilocal. One application generalizes a recently announced result of Picavet and Picavet-L'Hermitte on the minimal overrings of a local Noetherian ring. Examples are given to indicate sharpness of the results.