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Paper   IPM / M / 15701
School of Mathematics
  Title:   Chain conditions on commutative monoids
  Author(s):  Zahra Nazemian (Joint with B. Davvaz)
  Status:   Published
  Journal: Semigroup Forum
  Year:  2019
  Pages:   DOI: 10.1007/s00233-019-10032-1
  Supported by:  IPM
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 I1I2 ⊆ … of ideals of S, there exists n such that IiIn , as S-acts, for every in. 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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