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.