Chain conditions on commutative monoids

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.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Facchini, A., Nazemian, Z.: Modules with chain conditions up to isomorphism. J. Algebra 453, 578–601 (2016)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Facchini, A., Nazemian, Z.: Artinian dimension and isoradical of modules. J. Algebra 484, 66–87 (2017)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Rédei, L.: Theorie der endlich erzeugbaren kommutativen Halbgruppen. Physica, Würzburg (1963)

    Google Scholar 

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Zahra Nazemian.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by László Márki.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Davvaz, B., Nazemian, Z. Chain conditions on commutative monoids. Semigroup Forum 100, 732–742 (2020). https://doi.org/10.1007/s00233-019-10032-1

Download citation

Keywords

  • Semigroup
  • Monoid
  • Iso-DC
  • Iso-AC