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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Inertial subalgebras of central separable algebras
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by Nicholas S. Ford PDF
Proc. Amer. Math. Soc. 60 (1976), 39-44 Request permission

Abstract:

Let R be a commutative ring with 1. An R-separable subalgebra B of an R-algebra A is said to be an R-inertial subalgebra provided $B + N = A$, where N is the Jacobson radical of A. Suppose A is a finitely generated R-algebra which is separable over its center $Z(A)$. We show that if A possesses an R-inertial subalgebra B, then $Z(A)$ possesses a unique Rinertial subalgebra S. Moreover, A can be decomposed as $A \simeq B{ \otimes _S}Z(A)$. Suppose C is a finitely generated, commutative, semilocal R-algebra with Rinertial subalgebra S. We show that the R-inertial subalgebras of each central separable C-algebra are unique up to an inner automorphism generated by an element in the radical of the algebra if and only if the natural mapping of the Brauer groups $\beta (S) \to \beta (C)$ is a monomorphism. We conclude by presenting a method which enables one to construct algebras which possess nonisomorphic inertial subalgebras.
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 60 (1976), 39-44
  • MSC: Primary 16A16
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0414607-5
  • MathSciNet review: 0414607