Uniquely separable extensions

TL;DR

This paper investigates the properties of uniquely separable and split extensions of noncommutative rings, establishing conditions for their separability idempotents and connecting these concepts to depth and algebraic structures.

Contribution

It introduces the concept of uniquely separable extensions, characterizes their properties, and relates them to depth 1 and classical group-theoretic results.

Findings

Uniqueness of the separability idempotent characterizes depth 1 extensions.

A Frobenius extension with invertible E-index is uniquely separable if the centralizer equals the center.

Uniquely separable extensions of semisimple complex algebras have depth 1.

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.