Henselian pairs and Henselian local rings

Let $A$ be a commutative ring and let $I\subseteq A$ be an ideal.

The pair $(A,I)$ is a Henselian pair when:

1. $I\subseteq J(A)$, and
2. for every monic polynomial $f\in A[T]$ and every factorization $\bar{f}=g_0{h}_0$ in $(A\mathrm{/}I)[T]$ with $g_0,h_0$ monic and $(g_0,h_0)=(1)$, there is a unique factorization $f=gh$ in $A[T]$ with $g,h$ monic and $\bar{g}=g_0$, $\bar{h}=h_0$.

If $(A,\mathfrak{m})$ is a local ring, then $A$ is a Henselian local ring when the pair $(A,\mathfrak{m})$ is Henselian.

This page uses the Jacobson-radical condition as part of the definition rather than as a theorem proved later; that is the convention in the cited sources and is the hypothesis spent by the uniqueness and unit arguments below.