Korean J. Math. Vol. 29 No. 3 (2021) pp.483-492
DOI: https://doi.org/10.11568/kjm.2021.29.3.483

Reduced property over idempotents

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Published: Sep 30, 2021
Keywords:
reduced-over-idempotent ring, idempotent, reduced ring, Abelian ring, characteristic, unit, polynomial ring, power series ring
Mathematics Subject Classification:
16U40

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Tai Keun Kwak
Department of Mathematics, Daejin University, Pocheon 11159, Korea
Yang Lee
Department of Mathematics, Yanbian University, Yanji 133002, China and Institute of Basic Science, Daejin University, Pocheon 11159, Korea
Young Joo Seo
Department of Mathematics, Daejin University, Pocheon 11159, Korea

Abstract

This article concerns the property that for any element $a$ in a ring, if $a^{2n}=a^n$ for some $n\geq 2$ then $a^2=a$. The class of rings with this property is large, but there also exist many kinds of rings without that, for example, rings of characteristic $\neq 2$ and finite fields of characteristic $\geq 3$. Rings with such a property is called {\it 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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Supporting Agencies

National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT)

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