On splitting rings for Azumaya skew polynomial rings (Q2747689)

Let \(B\) be an associative ring with nonzero identity element, let \(\rho\) be an automorphism of \(B\) of finite order, say \(n\), and let \(B[x;\rho]\) be the skew polynomial ring in \(x\) over \(B\) such that \(1,x,x^2,\dots,x^{n-1}\) are independent over \(B\) and \(x^n\) is a unit of the ring \(B^\rho\) of all elements of \(B\) fixed under \(\rho\). Let \(\overline\rho\) be the inner automorphism of \(B[x;\rho]\) induced by \(x\), and assume that \(n\) is a unit in \(B\). The authors prove that for a \(\overline\rho\)-Galois extension \(B[x;\rho]\) over \((B[x;\rho])^{\overline\rho}\), \(B[x;\rho]\) is Azumaya if and only if \((B[x;\rho])^{\overline\rho}\) is Azumaya, and some splitting rings of \(B[x;\rho]\), \((B[x;\rho])^{\overline\rho}\) and \(B\) coincide.