Semi-simple and characteristic

Question

Let $C_3$ be the cyclic group of order 3, then let $K$ be a field, let $X=span_K(1+c+c^2)$, show that $X$ has a complement if and only if characteristic of $k$ is not equal to $3$.

Answer 1

Any direct summand of $K[C_3]$ is generated by an idempotent. In that case, $(1+c+c^2)R=eR$ where $e^2=e$.

If the characteristic of $K$ is $3$, then $(1+c+c^2)^2=0$, but that implies $((1+c+c^2)R)^2=0$. Thus whatever idempotent $e$ generates this summand satisfies $0=e^2=e$, but we know for a fact that $(1+c+c^2)R\neq 0$. So this contradicts the assumption it is a summand.

So if the characteristic is three, it cannot be a summand.

Possibly you already solved it this way in the other direction, but when the characteristic isn't $3$ you can show $e=\frac{1+c+c^2}{3}$ is idempotent, and $R=eR\oplus (1-e)R$.