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On the lattice of subalgebras of an algebra


Author: Linda L. Deneen
Journal: Proc. Amer. Math. Soc. 86 (1982), 189-195
MSC: Primary 16A16; Secondary 16A32
DOI: https://doi.org/10.1090/S0002-9939-1982-0667270-5
MathSciNet review: 667270
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Abstract: Let $ R$ be a Noetherian inertial coefficient ring and let $ A$ be a finitely generated $ R$-algebra (that is, finitely generated as an $ R$-module) with Jacobson radical $ J(A)$. Let $ S$ be a subalgebra of $ A$ with $ S + J(A) = A$. We show that for every separable subalgebra $ T$ of $ a$ there is a unit a of $ A$ such that $ aT{a^{ - 1}} \subseteq S$. It follows that if $ S$ is separable (hence inertial) and if $ T$ is a maximal separable subalgebra of $ A$, then $ T$ is inertial. We also show that if $ S + I = A$ for a nil ideal $ I$ of $ A$, then $ R$ can be taken to be an arbitrary commutative ring, and the conjugacy result still holds.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1982-0667270-5
Keywords: Separable algebra, inertial subalgebra, inertial coefficient ring, lifting idempotents
Article copyright: © Copyright 1982 American Mathematical Society

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