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Reflexive primes, localization and primary decomposition in maximal orders


Authors: J. H. Cozzens and F. L. Sandomierski
Journal: Proc. Amer. Math. Soc. 58 (1976), 44-50
MSC: Primary 16A08
DOI: https://doi.org/10.1090/S0002-9939-1976-0419494-7
MathSciNet review: 0419494
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Abstract: If $ R$ is a maximal order and $ P$ a reflexive prime ideal of $ R$, then the Goldie localization of $ R$ at $ P$ is shown to be the classical (partial) quotient ring of $ R$ with respect to the Ore set $ C(P) = \{ r \in R\vert rx \in P \Rightarrow x \in P\} $. This is accomplished by introducing new symbolic powers of the prime $ P$ which agree with Goldie's symbolic powers. As a consequence, whenever $ P$ is a reflexive prime ideal of $ R$ and $ {P^{(n)}}$ the $ n$th (Goldie) symbolic power of $ P$, then an ideal $ B$ is reflexive if and only if $ B = \bigcap\nolimits_{i = 1}^n {P_i^{({n_i})}} $ for uniquely determined reflexive primes $ {P_i}$ and integers $ {n_i} > 0$. More generally, each bounded essential right (left) ideal is shown to have a reduced primary decomposition and an explicit determination of the components is given in terms of the bound of the ideal.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1976-0419494-7
Keywords: Order, partial quotient ring, Ore set, equivalent order, maximal order, Asano order, reflexive module, symbolic power, associated prime, primary decomposition
Article copyright: © Copyright 1976 American Mathematical Society

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