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Semistar-super potent domains

  • Gmiza Wafa
  • Hizem Sana
Original Paper
  • 3 Downloads

Abstract

Let \(\star \) be a finite type star operation defined on an integral domain D. Call D \(\star \)-potent (resp., \(\star \)-super potent) if every maximal \(\star \)-ideal M of D containes a nonzero finitely generated ideal I such that I is contained in no other maximal \(\star \)-ideal of D (resp., I is contained in no other maximal \(\star \)-ideal and every finitely generated ideal J containing I is \(\star \)-invertible). In this paper, we generalize the notion of super potent domains in the context of semistar operations. We prove that if \(\star \) is a semistar operation on an integral domain D,  then D is \(\widetilde{\star }\)-(super) potent if and only if D[X] is \([\star ]\)-(super) potent if and only if the Nagata ring associated to \(\star \) is (super) potent, where \([\star ]\) is the semistar operation introduced by Chang and Fontana (J Algebra 318(1):484–493, 2007).

Keywords

Semistar operation Star operation Super potent domain Polynomial ring Nagata ring 

Mathematics Subject Classification

13F20 13G05 13A15 13E99 

Notes

Acknowledgements

The authors would like to thank the refree for his/her valuable comments.

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Copyright information

© The Managing Editors 2018

Authors and Affiliations

  1. 1.Department of MathematicsFaculty of SciencesMonastirTunisia

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