Daily Archives: November 1, 2013

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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