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