A generalization of Hensel lemma

Question

Let A be a noetherian ring, let I be a proper ideal of A. Suppose that A is complete and separate for the I-adic topology. Let F(X) ∈ A[X] be a polynomial such that there exists a ∈ A with F(a) ∈ I and (F'(a), I) = A. I want to show that there exists an actual root α ∈ A of F which moreover coincides with a mod I. I think this generalizes the classical Hensel Lemma over the ring of p-adic integers Zp

Answer 1

You can see such a result as Theorem 10.5 here. Replace $(F'(a),I) = A$ with $F'(a) \in A^\times$, which in practice may be equivalent.