On the Noetherian dimension of Artinian modules with homogeneous uniserial dimension

Document Type : Research Paper

Authors

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

Abstract

 ‎In this article‎, ‎we first‎ ‎show that non-Noetherian Artinian uniserial modules over‎ ‎commutative rings‎, ‎duo rings‎, ‎finite $R$-algebras and right‎ ‎Noetherian rings are $1$-atomic exactly like $\Bbb Z_{p^{\infty}}$‎. ‎Consequently‎, ‎we show that if $R$ is a right duo (or‎, ‎a right‎ ‎Noetherian) ring‎, ‎then the Noetherian dimension of an Artinian‎ ‎module with homogeneous uniserial dimension is less than or equal‎ ‎to $1$‎. ‎In particular‎, ‎if $A$ is a quotient finite dimensional‎ ‎$R$-module with homogeneous uniserial dimension‎, ‎where $R$ is a‎ ‎locally Noetherian (or‎, ‎a Noetherian duo) ring‎, ‎then $n$-dim ‎$A\leq‎ ‎1$‎. ‎We also show that the Krull dimension of Noetherian modules is‎ ‎bounded by the uniserial dimension of these modules‎. ‎Moreover‎, ‎we introduce the concept of qu-uniserial modules and by using this‎ ‎concept‎, ‎we observe that if $A$ is an Artinian $R$-module‎, ‎such that‎ ‎any of its submodules is qu-uniserial‎, ‎where $R$ is a right duo (or‎, ‎a right Noetherian) ring‎, ‎then $n$-dim $‎A\leq 1$.

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References

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Bulletin of the Iranian Mathematical Society
Volume 43, Issue 7
December 2017
Pages 2457-2470

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History

  • Receive Date: 15 January 2017
  • Revise Date: 09 September 2017
  • Accept Date: 09 September 2017

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