ON MINIMAL RING EXTENSIONS OF FINITE RINGS.

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, "either (i) or (ii)" is necessary and sufficient for R ⊂ S to be a minimal ring extension; and 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. Let R ⊂ Sj 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 (ramified, decomposed or inert) of minimal extension of R ↔ |Z(S₁)| = |Z(S₂)| ↔ |U(S₁)| = |U(S₂). [ABSTRACT FROM AUTHOR]