The role of the environment on the formation of S0 galaxies is still not well understood, specifically in the outskirts of galaxy clusters. We study eight low-redshift clusters, analyzing galaxy members up to cluster-centric distances $sim2.5,R_{200}$. We perform 2D photometric bulge-disk decomposition in the $g$-, $r$- and $i$-bands from which we identify 469 double-component galaxies. We analyze separately the colors of the bulges and the disks and their dependence on the projected cluster-centric distance and on the local galaxy density. For our sample of cluster S0 galaxies, we find that bulges are redder than their surrounding disks, show a significant color-magnitude trend, and have colors that do not correlate with environment metrics. On the other hand, the disks associated with our cluster S0s become significantly bluer with increasing cluster-centric radius, but show no evidence for a color-magnitude relation. The disk color-radius relation is mainly driven by galaxies in the cluster core at $0leq R/ R_{200}<0.5$. No significant difference is found for the disk colors of backsplash and infalling galaxies in the projected phase space. Beyond $R_{200}$, the disk colors do not change with the local galaxy density, indicating that the colors of double-component galaxies are not affected by pre-processing. A significant color-density relation is observed for single-component disk-dominated galaxies beyond $R_{200}$. We conclude that the formation of cluster S0 galaxies is primarily driven by cluster core processes acting on the disks, while evidence of pre-processing is found for single-component disk-dominated galaxies. We publicly release the data from the bulge-disk decomposition.
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