Let $a \in R$ and let $a^n$ be right quasi-regular, where n is some positive integer. Show that $a$ is right quasi-regular

Question

I have tried the following way: $(a^n◦x)=0$ so $(aaaa...a◦x)$ so $(aaaa...a)(a◦x)=0$ so $aaaa...a=0$ or $a◦x=0$ if $aaaa...a=0$ then $a=0$ and $(a◦x)=0$

Answer 1

By this post, there is an isomorphism $\varphi:(R,\circ)\to (R,\cdot), \: \varphi(a)=1-a$. So $a^n$ is right quasi-regular iff $1-a^n$ is left unit, i.e. there $x\in R$ that $(1-a^n)x=1$. But there is $$ (1-a^n)x=(1-a)(1+a+\cdots+a^{n-1})x=1 $$ Thus $1-a$ is also left unit and so $a$ is right quasi-regular.