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Proceedings of the American Mathematical Society

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



Inertial subalgebras of central separable algebras

Author: Nicholas S. Ford
Journal: Proc. Amer. Math. Soc. 60 (1976), 39-44
MSC: Primary 16A16
MathSciNet review: 0414607
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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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Keywords: Inertial subalgebra, separable algebra, Jacobson radical, Brauer group
Article copyright: © Copyright 1976 American Mathematical Society