Characterizing finite fields via minimal ring extensions

Abstract

Let A be a finite local (commutative unital) ring. If A is not a field, there exists a positive integer N such that for each ring B such that is an inert (minimal ring) extension. It follows that A is a (finite) field the inert extensions of A form infinitely many (equivalently, denumerably many) A-algebra isomorphism classes the minimal ring extensions of A form infinitely many (equivalently, denumerably many) A-algebra isomorphism classes there exists a minimal ring extension such that B is a flat A-module there exists a minimal ring extension such that B is a local ring and is a Galois ring extension.