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Finiteness properties of local cohomology modules for $ \mathfrak{a}$-minimax modules


Authors: Jafar Azami, Reza Naghipour and Bahram Vakili
Journal: Proc. Amer. Math. Soc. 137 (2009), 439-448
MSC (2000): Primary 13D45, 14B15, 13E05
DOI: https://doi.org/10.1090/S0002-9939-08-09530-0
Published electronically: August 25, 2008
MathSciNet review: 2448562
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Abstract: Let $ R$ be a commutative Noetherian ring and $ \mathfrak{a}$ an ideal of $ R$. In this paper we introduce the concept of $ \mathfrak{a}$-minimax $ R$-modules, and it is shown that if $ M$ is an $ \mathfrak{a}$-minimax $ R$-module and $ t$ a non-negative integer such that $ {\rm H}_\mathfrak{a}^i(M)$ is $ \mathfrak{a}$-minimax for all $ i<t$, then for any $ \mathfrak{a}$-minimax submodule $ N$ of $ {\rm H}_\mathfrak{a}^t(M)$, the $ R$-module $ {\rm Hom}_R(R/\mathfrak{a},{\rm H}_\mathfrak{a}^t(M)/N)$ is $ \mathfrak{a}$-minimax. As a consequence, it follows that the Goldie dimension of $ {\rm H}_\mathfrak{a}^t(M)/N$ is finite, and so the associated primes of $ {\rm H}_\mathfrak{a}^t(M)/N$ are finite. This generalizes the main result of Brodmann and Lashgari (2000).


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Additional Information

Jafar Azami
Affiliation: Department of Mathematics, University of Tabriz, Tabriz 51666-16471, Iran – and – Department of Mathematics, Mohaghegh Ardabily University, Ardabil, Iran
Email: azami@tabrizu.ac.ir

Reza Naghipour
Affiliation: Department of Mathematics, University of Tabriz, Tabriz 51666-16471, Iran – and – School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O. Box 19395-5746, Tehran, Iran
Email: naghipour@ipm.ir, naghipour@tabrizu.ac.ir

Bahram Vakili
Affiliation: Department of Mathematics, Science and Research Branch, Islamic Azad University, P.O. Box 14515-775, Tehran, Iran – and – Department of Mathematics, Shabestar Islamic Azad University, Shabestar, Iran
Email: bvakil@iaushab.ac.ir

DOI: https://doi.org/10.1090/S0002-9939-08-09530-0
Keywords: Goldie dimension, $\mathfrak a$-minimax modules, $\mathfrak a$-cominimax modules, local cohomology, associated primes.
Received by editor(s): October 3, 2007
Received by editor(s) in revised form: January 18, 2008
Published electronically: August 25, 2008
Additional Notes: The research of the second author was supported in part by a grant from IPM (No. 86130031)
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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