Separable equivalence of rings and symmetric algebras

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.