Modules with Chain Condition on Non-finitely Generated Submodules

被引：0
作者：
Maryam Davoudian
机构：
[1] Shahid Chamran University of Ahvaz,Department of mathematics
来源：
Mediterranean Journal of Mathematics | 2018年 / 15卷
关键词：
Non-finitely generated module; quotient finite dimensional module; Krull dimension; Noetherian dimension; 16P60; Secondary: 16P20; 16P40;
D O I：
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摘要：
In this article, we study modules with chain condition on non-finitely generated submodules. We show that if an R-module M satisfies the ascending chain condition on non-finitely generated submodules, then M has Noetherian dimension and its Noetherian dimension is less than or equal to one. In particular, we observe that if an R-module M satisfies the ascending chain condition on non-finitely generated submodules, then every submodule of M is countably generated. We investigate that if an R-module M satisfies the descending chain condition on non-finitely generated submodules, then M has Krull dimension and its Krull dimension may be any ordinal number α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}. In particular, if a perfect R-module M satisfies the descending chain condition on non-finitely generated submodules, then it is Artinian.
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