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|Paper IPM / M / 15701||
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 ⊆ … of ideals of S,
there exists n such that Ii ≅ In , as S-acts, for every 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 = S1 ×…×Sn is a finite product of monids such that
each Si 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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