Uniquely separable extensions

Abstract

The separability tensor element of a separable extension of noncommutative rings is an idempotent when viewed in the correct endomorphism ring; so one speaks of a separability idempotent, as one usually does for separable algebras. It is proven that this idempotent is full if and only the H-depth is 1 (H-separable extension). Similarly, a split extension has a bimodule projection; this idempotent is full if and only if the ring extension has depth 1 (centrally projective extension). Separable and split extensions have separability idempotents and bimodule projections in 1 - 1 correspondence via an endomorphism ring theorem in Section~3. If the separable idempotent is unique, then the separable extension is called uniquely separable. A Frobenius extension with invertible $E$-index is uniquely separable if the centralizer equals the center of the over-ring. It is also shown that a uniquely separable extension of semisimple complex algebras with invertible E-index has depth 1. Earlier group-theoretic results are recovered and related to depth $1$. The dual notion, uniquely split extension, only occurs trivially for finite group algebra extensions over complex numbers.