2020/05/14 by Kumar, Rahul, Gaur, Atul
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2005.07217
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 \hookrightarrow 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 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/\mathfrakm where \mathfrakm∈ 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.