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

Main Article Content

Tai Keun Kwak
Yang Lee
Young Joo Seo


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.

Article Details

Supporting Agencies

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


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