Completely Irreducible Submodules and Characterization of Distributive and Artinian Modules

Document Type : Original Paper

Author

Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran

Abstract

 
Let R be a commutative ring with identity and let M be a unitary R-module.In this paper, the structure of completely irreducible submodules will be studied and it is proved that a submodule K has a comlpetely irreducible divisor if and only if Soc(M/K) is nontrivial which implies that a maximal ideal m is an strongly Bourbaki associated prime ideal of K if and only if K has an m-primal completely irrducible divisor. Submodules of M that are representable as an irredundant intersection of an overfamily of completely irreducible submodules are characterized. Then it will be shown that, if R is a Noetherian ring, then M is Artinian if and only if its zero submodule has a primary decomposition whose components are completely irreducible submodules. Finally, it is proved that M is distributive if and only if the set of its completely irreducible submodules is: 
{ m(Rx)(m) | x ∈ M , m ∈ Max(R) ∩ Supp(Rx) }.
    

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References

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Journal of Advanced Mathematical Modeling
Volume 13, Issue 3 - Serial Number 31
December 2023
Pages 421-434

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History

  • Receive Date: 12 July 2023
  • Revise Date: 12 October 2023
  • Accept Date: 01 December 2023

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