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 Δ:R↪R×R, given by Δ(r)=(r,r) for all r∈R. It is shown that, for any a,b∈R, the extension Δ(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.