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The center of an order with finite global dimension


Author: Mark Ramras
Journal: Trans. Amer. Math. Soc. 210 (1975), 249-257
MSC: Primary 16A60
DOI: https://doi.org/10.1090/S0002-9947-1975-0374191-5
MathSciNet review: 0374191
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Abstract: Let $ \Lambda $ be a quasi-local ring of global dimension $ n < \infty $. Assume that its center $ R$ is a noetherian domain, that $ \Lambda $ is finitely generated torsion-free as an $ R$-module, and that $ R$ is an $ R$-direct summand of $ \Lambda $. Then $ R$ is integrally closed in its quotient field $ K$ and Macauley of dimension $ n$. Furthermore, when $ n = 2,\Lambda $ is a maximal $ R$-order in the central simple $ K$-algebra $ \Lambda { \otimes _R}K$. This extends an earlier result of the author, in which $ R$ was assumed to have global dimension 2. Examples are given to show that in the above situation $ R$ can have infinite global dimension.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0374191-5
Keywords: Maximal order, global dimension, center, Macauley ring, integrally closed, skew power series ring
Article copyright: © Copyright 1975 American Mathematical Society

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