IMC 2000 · Problem 5

Let $R$ be a ring of characteristic zero (not necessarily commutative). Let $e$, $f$ and $g$ be idempotent elements of $R$ satisfying $e + f + g = 0$. Show that $e = f = g = 0$.

($R$ is of characteristic zero means that, if $a \in R$ and $n$ is a positive integer, then $na \ne 0$ unless $a = 0$. An idempotent $x$ is an element satisfying $x = x^2$.)