Definition: An element in a ring is called central if is in the center of i.e. for all

1) Suppose is a ring in which Let be an idempotent, i.e. Prove that is central.

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2) Use 1) to give a short proof of this very special case of Jacobson's theorem: if for all in a ring with identity then is commutative.

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3) This time suppose for all in a ring with identity Use 1) to prove that is commutative.

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4)Challenge: Suppose for all in a ring with identity Use 1) to prove that is commutative.