On minimal ring extensions

Abstract

Let R be a commutative ring with identity. The ring R × R can be viewed as an extension of R via the diagonal map : R →֒ R×R, given by (r) = (r, r) for all r ∈ R. It is shown that, for any a, b ∈ R, the extension (R)[(a, b)] ⊂ R×R is a minimal ring extension if and only if the ideal < a−b > is a maximal ideal of R. A complete classification of maximal subrings of R(+)R is also given. The minimal ring extension of a von Neumann regular ring R is either a von Neumann regular ring or the idealization R(+)R/m where m ∈ Max(R). If R ⊂ T is a minimal ring extension and T is an integral domain, then (R : T) = 0 if and only if R is a field and T is a minimal field extension of R, or RJ is a valuation ring of altitude one and TJ is its quotient field.