Let $\displaystyle R=\bigoplus_{g\in G}R_{g} $ be a graded ring. If $a$ is a homogeneous idempotent, then $\operatorname{degr}(a)=e$, where $e$ is the neutral element.
Since $a$ is a homogeneous element, there is $g\in G$, such that $a\in R_{g}$, then $a=aa\in R_{g}R_{g}\subseteq R_{g^{2}}$, but $\displaystyle R=\bigoplus_{g\in G}R_{g} $, then $g=g^{2}$? how could I conclude that $g=e$?