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 \( a^2 = a \). )

The best solution was submitted by 박훈민. Congratulations!

Similar solutions are submitted by 김동현(+3), 김호진(+3), 도수일(+3), 박민재(+3), 정성진(+3), 진우영(+3). Thank you for your participation.

Remark: Special thanks to 김동현, who first reported that the condition `characteristic zero’ is insufficient for the problem.