Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Maximal orders and reflexive modules

Author: J. H. Cozzens
Journal: Trans. Amer. Math. Soc. 219 (1976), 323-336
MSC: Primary 16A18
MathSciNet review: 0419503
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Abstract: If R is a maximal two-sided order in a semisimple ring and $ {M_R}$ is a finite dimensional torsionless faithful R-module, we show that $ m = {\text{End}_R}\;{M^\ast}$ is a maximal order. As a consequence, we obtain the equivalence of the following when $ {M_R}$ is a generator:

1. M is R-reflexive.

2. $ k = {\text{End}}\;{M_R}$ is a maximal order.

3. $ k = {\text{End}_R}\;{M^\ast}$ where $ {M^\ast} = {\hom _R}(M,R)$.

When R is a prime maximal right order, we show that the endomorphism ring of any finite dimensional, reflexive module is a maximal order. We then show by example that R being a maximal order is not a property preserved by k. However, we show that $ k = {\text{End}}\;{M_R}$ is a maximal order whenever $ {M_R}$ is a maximal uniform right ideal of R, thereby sharpening Faith's representation theorem for maximal two-sided orders. In the final section, we show by example that even if $ R = {\text{End}_k}V$ is a simple pli (pri)-domain, k can have any prescribed right global dimension $ \geqslant 1$, can be right but not left Noetherian or neither right nor left Noetherian.

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Keywords: Order, equivalent order, maximal order, Goldie ring, Ore domain, principal left ideal ring, simple ring, torsionless module, reflexive module, projective module, generator, uniform module
Article copyright: © Copyright 1976 American Mathematical Society