Reduced property over idempotents

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.