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Dimension of Non-small Submodules

  • Maryam DavoudianEmail author
Original Paper
  • 2 Downloads

Abstract

In this article, we introduce and study the concepts of non-small Krull dimension and non-small Noetherian dimension of an R-module M, where R is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension of modules. They, respectively, rely on the behavior of descending and ascending chains of non-small submodules. We show that, if the non-small Noetherian dimension (resp., non-small Krull dimension) of an R-module M equals \(\alpha \), then \(\frac{M}{\mathrm{Rad}(M)}\) has Noetherian dimension (resp., Krull dimension) and its Noetherian dimension (resp., Krull dimension) is less than or equal to \(\alpha \) (resp., \( \alpha +1\)). For decomposable modules, we show that the non-small Krull dimension (resp., non-small Noetherian dimension) and the Krull dimension (resp., Noetherian dimension) coincide.

Keywords

Noetherian dimension Small submodule Krull dimension Non-small Krull dimension Non-small Noetherian dimension 

Mathematics subject classification

16P60 16P20 16P40 

Notes

Acknowledgements

The author would like to thank professor O.A.S. Karamzadeh for a useful discussion and his encouragement during the preparation of a revised version of this article. She would also like to thank the referee for carefully reading the paper, giving a detailed report with very helpful comments.

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Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Department of MathematicsShahid Chamran University of AhvazAhvazIran

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