# 2013-18 Idempotent elements

Let $$R$$ be a ring of characteristic zero. Assume further that $$na \neq 0$$ for a positive integer $$n$$ and $$a \in R$$ unless $$a = 0$$. Suppose that $$e, f, g \in R$$ are idempotent (with respect to the multiplication) and satisfy $$e + f + g = 0$$. Show that $$e = f = g = 0$$. (An element $$a$$ is idempotent if $$a^2 = a$$. )

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