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

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    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).

    Thanks in advance.
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    Quote Originally Posted by Biscaim View Post
    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(1e) must be 0, so ey=eye. The same argument with e and 1e exchanged tells you that ye=eye. Hence ey=ye.
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