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Proceedings of the American Mathematical Society

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

 

 

Krull dimension and reflexivity in some Noetherian rings


Authors: A. Haghany and B. Sarath
Journal: Proc. Amer. Math. Soc. 83 (1981), 1-7
MSC: Primary 16A08; Secondary 16A33, 16A55
DOI: https://doi.org/10.1090/S0002-9939-1981-0619968-1
MathSciNet review: 619968
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Abstract: In this paper we study Noetherian prime rings $ R$ which satisfy the formula $ \left\vert R \right\vert = \sup \{ \left\vert {{I^{* *}}/I} \right\vert:I$ is an essential left ideal of $ R\} + 2$, where $ \vert\;\vert$ denotes left Krull dimension. If further $ Q/R$ is $ \left\vert R \right\vert - 1$ unmixed, where $ Q$ is the simple Artinian quotient ring of $ R$, we characterize $ R$ using torsion theories cogenerated by the injective hulls of $ \left\vert R \right\vert - 1$ dimensional critical modules. Also equivalent statements are established, linking homological properties with dimension theory, for $ R$-modules to be reflexive.


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DOI: https://doi.org/10.1090/S0002-9939-1981-0619968-1
Keywords: Reflexive module, Krull dimension, torsion theory
Article copyright: © Copyright 1981 American Mathematical Society