  We study a systematic way to produce a Lipschitz operator ideal from a Banach
linear operator ideal $\mathcal A$. For maximal and minimal operator ideals
$\mathcal A$, the Lipschitz maximal hull and minimal kernel of the Lipschitz
operator ideals $\text{Lip}_0 \circ \mathcal A \circ \text{Lip}_0$ are
investigated, respectively. Using ultraproduct techniques, we obtain the
Lipschitz version of a result of K\"ursten and Piestch which characterizes the
maximal hull of any Lipschitz operator ideal. Among other results, we
characterize the linear operators which belong to $\text{Lip}_0\circ \mathcal
A\circ \text{Lip}_0$ which, in many cases, they are precisely those which are
in $\mathcal A$. In particular, we give some cases in which a nonlinear
factorization of linear operators implies a linear one, in terms of a given
Banach linear operator ideal $\mathcal A$.
