On separable subalgebras of Azumaya algebras

Abstract

Let $A$ be an Azumaya $C$-algebra. Then the set of all commutative separable subalgebras of $A$ and the set of separable subalgebras $B$ such that $V_A(B)=V_B(B)$ are in a one-to-one correspondence, where $V_A(B)$ is the commutator subring of $B$ in $A$, and the set of all separable subalgebras of $A$ is a disjoint union of the Azumaya algebras in $A$ over a commutative separable subalgebra of~$A$. The results are used to compute splitting rings for an Azumaya skew group ring.