Semistar-super potent domains

  • Gmiza WafaEmail author
  • Hizem Sana
Original Paper


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).


Semistar operation Star operation Super potent domain Polynomial ring Nagata ring 

Mathematics Subject Classification

13F20 13G05 13A15 13E99 



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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