Chain conditions on commutative monoids

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 I1⊆I2⊆⋯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_1 \subseteq I_2 \subseteq \cdots $$\end{document} of ideals of S, there exists n such that Ii≅In\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_i \cong I_n $$\end{document}, as S-acts, for every i≥n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i \ge n$$\end{document}. 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=S1×⋯×Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S= S_1 \times \cdots \times S_n$$\end{document} is a finite product of monids such that each Si\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_i$$\end{document} is isosimple, then S may not be Iso-DC but it is a noetherian S-act and so an Iso-AC monoid.