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Azumaya algebras and skew polynomial rings. II. (English) Zbl 0565.16002
The author continues his study of Azumaya algebras and skew polynomial rings [ibid. 23, 19-32 (1981; Zbl 0475.16002)]. He establishes sufficient conditions for the skew Laurent polynomial ring \(B[X,X^{-1};\rho]\) to be an Azumaya algebra over a suitable central subring, where B is an Azumaya C-algebra and \(\rho\) is an automorphism of C. He also considers questions of separability for the normal monic elements of B[X;\(\rho\) ].
Let \(\sigma\) denote the restriction of \(\rho\) to C, G be the cyclic group generated by \(\sigma\), m be the order of G and Q be the subring of C consisting of those elements fixed by \(\sigma\). Suppose that \(2\leq m<\infty\) and that C is a G-Galois extension of Q. The main result of the paper is that if there exists an invertible element u of B such that \(\rho (u)=u\) and \(\rho^ m(a)=u^{-1}au\) for all \(a\in B\), then \(B[X,X^{-1};\rho]\) is an Azumaya algebra over \(Q[X^ mu^{-1},X^{- 1}u].\)
This result is applied to the case where C is \(\sigma\)-prime, that is IJ\(\neq 0\) for all non-zero ideals I, J of C such that \(\sigma\) (I)\(\subseteq I\) and \(\sigma\) (J)\(\subseteq J\). In this case Q is a domain and has a quotient field F. If there exists an invertible element u of B such that \(\rho (u)=u\) and \(\rho^ m(a)=u^{-1}au\) for all \(a\in B\), then \((B\otimes_ QF)[X,X^{-1};\rho \otimes 1_ F]\) is an Azumaya algebra over \(F[X^ mu^{-1},X^{-m}u]\). This result had been proved for the case \(B=C\) by R. S. Irving [J. Algebra 56, 315-342 (1979; Zbl 0399.16015)].
Reviewer: D.A.Jordan

MSC:
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16W20 Automorphisms and endomorphisms
16S20 Centralizing and normalizing extensions
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