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Title:Uniform dimension of modules
Authors: Roy McCasland ; Patrick Smith
Date:Dec 2004
Publication Title:The Quarterly Journal of Mathematics
Publisher:OUP
Publication Type:Journal Article Publication Status:Published
Volume No:55(4) Page Nos:491-498
DOI:10.1093/qmath/hah007
Abstract:
Let $M$ be a module which has finite uniform dimension and let $K_i(1 \leq i \leq n)$ be a finite collection of submodules of $M$ such that $0 = K_1 \cap \cdots \cap K_n$. Then the uniform dimension $u(M)$ of $M$ is the sum of the uniform dimensions of the factor modules $M/K_i(1 \leq i \leq n)$ if and only if $K_i$ is a complement of $K_1 \cap \cdots \cap K_{i-1} \cap K_{i+1} \cap \cdots \cap K_n$ in $M$ for each $1 \leq i \leq n$. In case $K_i$ is $P_i$-prime for some prime ideal $P_i$ for each $1 \leq i \leq n$, the prime ideals $P_i\;(1 \leq i \leq n)$ are distinct and $0 \neq K_1 \cap \cdots \cap K_{i-1} \cap K_{i+1} \cap \cdots \cap K_n$ for each $1 \leq i \leq n$, then it is shown that $u(M) = \sum^n_{i=1}\;u(L_i/(L_i \cap K_i))$ for certain submodules $L_i(1 \leq i \leq n)$ of $M$.
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Bibtex format
@Article{EDI-INF-RR-0461,
author = { Roy McCasland and Patrick Smith },
title = {Uniform dimension of modules},
journal = {The Quarterly Journal of Mathematics},
publisher = {OUP},
year = 2004,
month = {Dec},
volume = {55(4)},
pages = {491-498},
doi = {10.1093/qmath/hah007},
url = {http://qjmath.oxfordjournals.org/content/vol55/issue4/index.dtl},
}

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