Inertial subalgebras of central separable algebras

Ford, Nicholas S.

doi:10.1090/S0002-9939-1976-0414607-5

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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