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Chain conditions on commutative monoids

  • Bijan Davvaz
  • Zahra NazemianEmail author
Research Article
  • 15 Downloads

Abstract

We consider commutative monoids with some kinds of isomorphism condition on their ideals. We say that a monoid S has isomorphism condition on its ascending chains of ideals, if for every ascending chain \(I_1 \subseteq I_2 \subseteq \cdots \) of ideals of S, there exists n such that \(I_i \cong I_n \), as S-acts, for every \(i \ge n\). Then S for short is called Iso-AC monoid. Dually, the concept of Iso-DC is defined for monoids by isomorphism condition on descending chains of ideals. We prove that if a monoid S is Iso-DC, then it has only finitely many non-isomorphic isosimple ideals and the union of all isosimple ideals is an essential ideal of S. If a monoid S is Iso-AC or a reduced Iso-DC, then it cannot contain a zero-disjoint union of infinitely many non-zero ideals. If \(S= S_1 \times \cdots \times S_n\) is a finite product of monids such that each \(S_i\) is isosimple, then S may not be Iso-DC but it is a noetherian S-act and so an Iso-AC monoid.

Keywords

Semigroup Monoid Iso-DC Iso-AC 

Notes

Acknowledgements

The authors would like to thank the referee for evaluation and improving some results such as Lemma 2.1, Proposition 2.3, Theorem 3.11 and Corollary 3.13. The authors thank Iran National Science Foundation (INSF) and Yazd University for their support through the grant no. 94015014. The second author was also supported by a grant from IPM.

References

  1. 1.
    Facchini, A., Nazemian, Z.: Modules with chain conditions up to isomorphism. J. Algebra 453, 578–601 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Facchini, A., Nazemian, Z.: Artinian dimension and isoradical of modules. J. Algebra 484, 66–87 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dickson, L.E.: Finiteness of the odd perfect and primitive abundant numbers with \(n\) distinct prime factors. Am. J. Math. 35, 413–422 (1913)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Rédei, L.: Theorie der endlich erzeugbaren kommutativen Halbgruppen. Physica, Würzburg (1963)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsYazd UniversityYazdIran
  2. 2.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran

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