1. Proof Problem (Rings): Let R be a ring. An idempotent of R is an element a e R such that a? is the multiplicative identity, 1 (if R is a ring with unity). Suppose R is a ring with unity and a E R is an idempotent element. Prove the following: = a. An example of an idempotent element (a) a(1 – a) is an idempotent element. (b) 1- a is an idempotent element. (c) 2a – 1 is invertible. That is, there exists an element x E R such that x(2a – 1) = 1. -

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