Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
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REDUCED PROPERTY OVER IDEMPOTENTS

  • Kwak, Tai Keun (Department of Mathematics, Daejin University) ;
  • Lee, Yang (Department of Mathematics, Yanbian University and Institute of Basic Science, Daejin University) ;
  • Seo, Young Joo (Department of Mathematics, Daejin University)
  • Received : 2021.02.06
  • Accepted : 2021.07.11
  • Published : 2021.09.30
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Abstract

This article concerns the property that for any element a in a ring, if a2n = an for some n ≥ 2 then a2 = a. The class of rings with this property is large, but there also exist many kinds of rings without that, for example, rings of characteristic ≠2 and finite fields of characteristic ≥ 3. Rings with such a property is called reduced-over-idempotent. The study of reduced-over-idempotent rings is based on the fact that the characteristic is 2 and every nonzero non-identity element generates an infinite multiplicative semigroup without identity. It is proved that the reduced-over-idempotent property pass to polynomial rings, and we provide power series rings with a partial affirmative argument. It is also proved that every finitely generated subring of a locally finite reduced-over-idempotent ring is isomorphic to a finite direct product of copies of the prime field {0, 1}. A method to construct reduced-over-idempotent fields is also provided.

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Acknowledgement

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

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