| dc.description.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. |
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