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Paper IPM / M / 15701  


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} ⊆ I_{2} ⊆ … of ideals of S,
there exists n such that I_{i} ≅ I_{n} , as Sacts, for every i ≥ n. Then S for short is called
IsoAC monoid. Dually, the concept of IsoDC
is defined for monoids by isomorphism condition on descending chains of ideals.
We prove that if a monoid S is IsoDC,
then it has only finitely many nonisomorphic
isosimple ideals and the union of all isosimple ideals is an
essential ideal of S.
If a monoid S is
IsoAC or a reduced IsoDC, then it
cannot contain a
zerodisjoint union of infinitely many
nonzero ideals. If S = S_{1} ×…×S_{n} is a finite product of monids such that
each S_{i} is isosimple, then S may not be IsoDC but it is a noetherian Sact and so
an IsoAC monoid.
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