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

On the Lipschitz operator ideal $\text{Lip}_{0}\circ \mathcal A\circ \text{Lip}_{0}$