Modules with Chain Condition on Non-finitely Generated Submodules

Modules with Chain Condition on Non-finitely Generated Submodules 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 $$\alpha$$ α . In particular, if a perfect R-module M satisfies the descending chain condition on non-finitely generated submodules, then it is Artinian. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mediterranean Journal of Mathematics Springer Journals

Modules with Chain Condition on Non-finitely Generated Submodules

, Volume 15 (1) – Dec 1, 2017
12 pages

/lp/springer_journal/modules-with-chain-condition-on-non-finitely-generated-submodules-58Gsu6VHWl
Publisher
Springer International Publishing
Copyright © 2017 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Mathematics, general
ISSN
1660-5446
eISSN
1660-5454
D.O.I.
10.1007/s00009-017-1047-y
Publisher site
See Article on Publisher Site

Abstract

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 $$\alpha$$ α . In particular, if a perfect R-module M satisfies the descending chain condition on non-finitely generated submodules, then it is Artinian.

Journal

Mediterranean Journal of MathematicsSpringer Journals

Published: Dec 1, 2017

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