Azumaya algebras and skew polynomial rings. II.

*(English)*Zbl 0565.16002The 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)].

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 |