A closed ideal I of a commutative C^{∗}-algebra A and a closed ideal J of I such that J isn’t an ideal of A.

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Let A = C([0, 1]), I = Af and J = Cf + Af^{2},where f(t) = t; 0 ≤ t ≤ 1. Then J is an ideal of I and I is an ideal of A; but f ∈ J and f.f^{\frac{1}{2}} ∉ J (otherwise, there exist λ ∈ C and g ∈ A such that f.f^{\frac{1}{2}} = λf +gf^{2}. So lim_{t→0} t^{\frac{1}{2}} = λ+lim_{t→0} tg(t). Therefore λ = 0 and t^{\frac{1}{2}} = tg contradicting the continuity of g. Thus J isn’t an ideal of A.