Exercise 12, § 5

Let A be a ring and a a finite type ideal of A. a) Prove that if there exists an ideal b of A such that a = a.b, there exists b ∈ b such that (1 − b)a = (0). b) Prove that if a² = a, there exists an element a ∈ a such that a² = a and a = Aa. c) Suppose that a is maximal and that the A-module a/a² can be generated by n elements. Prove that the ideal a can be generated by n + 1 elements. (One takes elements x₁, ..., xₙ in a whose images in a/a² generate the latter. Denoting by x the ideal of A generated by {x₁, ..., xₙ}, one remarks that in A/x one has (a/x)² = (a/x).)