# Math Help - idempotent elements of R belongs to cent(R)

1. ## idempotent elements of R belongs to cent(R)

If $R$ is a ring which has no nonzero nilpotent elements, deduce that all idempotent elements of $R$ belongs to $cent(R)$.

If $R$ is a ring which has no nonzero nilpotent elements, deduce that all idempotent elements of $R$ belongs to $cent(R)$.
If $e,y\in R$ and e is idempotent then $\bigl(ey(1-e)\bigr)^2 = 0$. If there are no nozero nilpotent elements then ey(1–e) must be 0, so ey=eye. The same argument with e and 1–e exchanged tells you that ye=eye. Hence ey=ye.