Inertial subalgebras of central separable algebras

Author:
Nicholas S. Ford

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

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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 , where *N* is the Jacobson radical of *A*. Suppose *A* is a finitely generated *R*-algebra which is separable over its center . We show that if *A* possesses an *R*-inertial subalgebra *B*, then possesses a unique *R*inertial subalgebra *S*. Moreover, *A* can be decomposed as . Suppose *C* is a finitely generated, commutative, semilocal *R*-algebra with *R*inertial 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 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

DOI:
https://doi.org/10.1090/S0002-9939-1976-0414607-5

Keywords:
Inertial subalgebra,
separable algebra,
Jacobson radical,
Brauer group

Article copyright:
© Copyright 1976
American Mathematical Society