Let I ⊆ R be an ideal. (a) Prove that every element of R/I is a solution of x2 = x if and only if r2 − r ∈ I for all r ∈ R. Is R/I an integral domain? (b) Suppose that R is an integral domain. Is R/I necessarily an integral domain? If so, prove it. If not, provide a counterexample.

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