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 $$I_1 \subseteq I_2 \subseteq \cdots $$ I 1 ⊆ I 2 ⊆ ⋯ of ideals of S , there exists n such that $$I_i \cong I_n $$ I i ≅ I n , as S -acts, for every $$i \ge n$$ i ≥ 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 $$S= S_1 \times \cdots \times S_n$$ S = S 1 × ⋯ × S n is a finite product of monids such that each $$S_i$$ S i is isosimple, then S may not be Iso-DC but it is a noetherian S -act and so an Iso-AC monoid.

中文翻译：

---

可交换幺半群的链条件

我们考虑在其理想上具有某种同构条件的可交换幺半群。我们说幺半群 S 在其上升的理想链上具有同构条件，如果对于 S 的理想的每个上升链 $$I_1 \subseteq I_2 \subseteq \cdots $$ I 1 ⊆ I 2 ⊆ ⋯ ，存在 n 使得$$I_i \cong I_n $$ I i ≅ I n ，作为 S -acts，对于每一个 $$i \ge n$$ i ≥ n 。那么S简称为Iso-AC幺半群。双重地，Iso-DC 的概念是通过理想降链上的同构条件为幺半群定义的。我们证明，如果一个幺半群 S 是 Iso-DC，那么它只有有限多个非同构同构理想，并且所有同构理想的并集是 S 的本质理想。如果幺半群 S 是 Iso-AC 或约化的 Iso-DC，则它不能包含无限多个非零理想的零不相交并集。