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Proceedings of the American Mathematical Society

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

 
 

 

The dominion and separable subalgebras of finitely generated algebras


Author: Dean Sanders
Journal: Proc. Amer. Math. Soc. 48 (1975), 1-7
MSC: Primary 16A16
DOI: https://doi.org/10.1090/S0002-9939-1975-0364331-1
MathSciNet review: 0364331
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Abstract: Let $ R \subseteq S \subseteq A$ be rings with $ R$ commutative, $ A$ an $ R$-algebra which is finitely generated as an $ R$-module by $ n$-elements, and $ S$ a subalgebra of $ A$. Setting $ {D_0} = S$ and $ {D_i} = \operatorname{Dom} (R,{D_{i - 1}})$ for $ i \geqslant 1$, we show that $ {D_n} = R$. We use this to show that if $ S$ is a separable $ R$-algebra then $ S$ is also finitely generated as an $ R$-module. Finally, a characterization of $ S$ is given when $ A$ is commutative and $ S$ is an inertial subalgebra of $ A$.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0364331-1
Keywords: Dominion, epimorphism, separable subalgebra, inertial subalgebra
Article copyright: © Copyright 1975 American Mathematical Society

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