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On $$\alpha $$ α -Tall Modules

On $$\alpha $$ α -Tall Modules In this article, we introduce and study $$\alpha $$ α -tall modules. We show that an $$\alpha $$ α -tall module, where $$\alpha \ge 0$$ α ≥ 0 , is a tall module, i.e. M contains a submodule N such that N and $$\frac{M}{N}$$ M N are both non-Noetherian. We observe that every submodule of $$\alpha $$ α -tall modules is countably generated, where $$\alpha $$ α is countable. It is shown that if M is a $$\beta $$ β -atomic module, where $$\beta =\alpha +2$$ β = α + 2 , for some ordinal $$\alpha $$ α , then M is $$\alpha $$ α -tall. It is also proved that if M is an $$\alpha $$ α -atomic module, where $$\alpha $$ α is a limit ordinal, then M is both an $$\alpha $$ α -tall and $$\alpha $$ α -short module. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Malaysian Mathematical Sciences Society Springer Journals

On $$\alpha $$ α -Tall Modules

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References (30)

Publisher
Springer Journals
Copyright
Copyright © 2016 by Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia
Subject
Mathematics; Mathematics, general; Applications of Mathematics
ISSN
0126-6705
eISSN
2180-4206
DOI
10.1007/s40840-016-0422-3
Publisher site
See Article on Publisher Site

Abstract

In this article, we introduce and study $$\alpha $$ α -tall modules. We show that an $$\alpha $$ α -tall module, where $$\alpha \ge 0$$ α ≥ 0 , is a tall module, i.e. M contains a submodule N such that N and $$\frac{M}{N}$$ M N are both non-Noetherian. We observe that every submodule of $$\alpha $$ α -tall modules is countably generated, where $$\alpha $$ α is countable. It is shown that if M is a $$\beta $$ β -atomic module, where $$\beta =\alpha +2$$ β = α + 2 , for some ordinal $$\alpha $$ α , then M is $$\alpha $$ α -tall. It is also proved that if M is an $$\alpha $$ α -atomic module, where $$\alpha $$ α is a limit ordinal, then M is both an $$\alpha $$ α -tall and $$\alpha $$ α -short module.

Journal

Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals

Published: Sep 28, 2016

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