On minimal ring extensions of finite rings

Abstract

Two conditions, (i) and (ii), are defined, that may hold for a given (unital) ring extension R ⊂ S of (unital, associative, not necessarily commutative) finite rings. It is shown that if S is commutative, then ``"either (i) or (ii)” is a necessary and sufficient condition for R ⊂ S to be a minimal ring extension; and that for such extensions, (i) and (ii) are logically independent. For extensions with S (finite and) noncommutative, "either (i) or (ii)” is neither necessary nor sufficient for R ⊂ S to be a minimal ring extension; and for such minimal ring extensions, (i) and (ii) are logically independent. Next, let R ⊂ S_j be minimal ring extensions with S_j  (finite and) commutative (for j=1,2) and R local. Then: S₁ and S₂ are the same type (that is, ramified, decomposed or inert) of minimal extension of R ↔ |Z(S_1)|=|Z(S_2)| ↔ |U(S_1)|=|U(S_2)|.