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Dimension of Non-finitely Generated Submodules

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Abstract

In this article, we introduce and study the concepts of quasi-Krull dimension and quasi-Noetherian dimension of an R-module, where R is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension. They respectively rely on the behavior of descending and ascending chains of non-finitely generated submodules. It is proved that an R-module M has quasi-Krull dimension if and only if it has Krull dimension, but the values of these dimensions might differ. Similarly, an R-module M has quasi-Noetherian dimension if and only if it has Noetherian dimension. We also show that the Noetherian dimension of an R-module M with quasi-Noetherian dimension β is either β or β+1. In particular, if M is an α+1-atomic module, where α is a limit ordinal number, then the quasi-Noetherian dimension of M also is α+1.

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Acknowledgments

The author would like to thank the referee for carefully reading the paper, detailed report, and giving very helpful comments.

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  1. Department of Mathematics, Shahid Chamran University Of Ahvaz, Ahvaz, Iran

    Maryam Davoudian

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  1. Maryam Davoudian
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Correspondence to Maryam Davoudian.

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Davoudian, M. Dimension of Non-finitely Generated Submodules. Vietnam J. Math. 44, 817–827 (2016). https://doi.org/10.1007/s10013-016-0206-y

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  • DOI: https://doi.org/10.1007/s10013-016-0206-y

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