Separable equivalence of rings and symmetric algebras

TL;DR

This paper explores the structure of symmetric separable equivalent rings, showing they are connected via Frobenius bimodules and analyzing properties of bimodules over symmetric algebras, extending the theory of ring equivalences.

Contribution

It proves that symmetric separable equivalent rings are linked by Frobenius bimodules with specific separability properties and examines the nature of bimodules over symmetric algebras.

Findings

Symmetric separable equivalent rings are connected by Frobenius bimodules.

The ring extension from a symmetric algebra to its endomorphism ring is split and separable.

Bimodules over symmetric algebras are Frobenius, twisted by Nakayama automorphisms.

Abstract

We continue a study of separable equivalence from Hokkaido Mathematical Journal 24 (1995), 527-549. We prove that symmetric separable equivalent rings $A$ and $B$ are linked by a Frobenius bimodule ${}_AP_B$ such that $A$ is $P$-separable over $B$. Separably equivalent rings are linked by a biseparable bimodule $P$. In addition, the ring extension $A \rightarrow$ End $P_B$ is split, separable Frobenius. It is observed that left and right finite projective bimodules over symmetric algebras are Frobenius bimodules; twisted by the Nakayama automorphisms if over Frobenius algebras.