# Relation between lifts of simple roots and lifts of idempotents (Henselian property)

Let $$f:A\to B$$ be a morphism of commutative rings. Given a monic $$\varphi\in A[x]$$ write $$Z(A,\varphi)$$ for the set of simple roots of $$\varphi$$ in $$A$$. Consider the following properties of $$f:A\to B$$.

1. For each monic $$\varphi \in A[x]$$, there's an induced surjection (bijection) $$Z(A,\varphi)\longrightarrow Z(B,f(\varphi)).$$
2. For each monic $$\varphi\in A[x]$$ the boolean algebra morphism $$\mathrm{idemp}(A[x]/(\varphi))\to \mathrm{idemp}(B[x]/(f(\varphi)))$$ is surjective (bijective).
3. (Compare 09XI) For each integral base change the induced boolean algebra morphism between idempotents is surjective (bijective).

Are these properties equivalent? The motivation for my question comes from trying to get an overview of the notion of Henselian. There are many bits and pieces which I hope are underlain by equivalence of the above conditions. I am especially hopeful for "direct" proofs that go through as few other conditions as possible.

Forgive me if this question is too elementary for MO.