Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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The Relation Between Units and Nilpotents

  • Cheon, Jeoung Soo (Department of Mathematics, Pusan National University) ;
  • Kwak, Tai Keun (Department of Data Science, Daejin University) ;
  • Lee, Yang (Department of Mathematics, Yanbian University, Institute for Applied Mathematics and Optics, Hanbat National University) ;
  • Seo, Young Joo (Department of Data Science, Daejin University)
  • Received : 2021.11.17
  • Accepted : 2022.03.08
  • Published : 2022.06.30
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Abstract

We discuss the relation between units and nilpotents of a ring, concentrating on the transitivity of units on nilpotents under regular group actions. We first prove that for a ring R, if U(R) is right transitive on N(R), then Köthe's conjecture holds for R, where U(R) and N(R) are the group of all units and the set of all nilpotents in R, respectively. A ring is called right UN-transitive if it satisfies this transitivity, as a generalization, a ring is called unilpotent-IFP if aU(R) ⊆ N(R) for all a ∈ N(R). We study the structures of right UN-transitive and unilpotent-IFP rings in relation to radicals, NI rings, unit-IFP rings, matrix rings and polynomial rings.

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Acknowledgement

The authors thank the referee for very careful reading of the manuscript and many valuable suggestions that improved the paper by much.

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