A measurement of the ratio R c ¼ ð c ! J=c þ Þ=J=c in pC, pTi, and pW interactions at 920 GeV=c ( ffiffi ffi s p ¼ 41:6 GeV) in the Feynman-x range À0:35 < x J=c F < 0:15 is presented. Both þ À and e þ e À J=c decay channels are observed with an overall statistics of about 15 000 c events, which is by far the largest available sample in pA collisions. The result is R c ¼ 0:188 AE 0:013 st þ0:024 À0:022sys averaged over the different materials, when no J=c and c polarizations are considered. The c1 to c2 production ratio R 12 ¼ R c 1 =R c 2 is measured to be 1:02 AE 0:40, leading to a cross section ratio ð c1 Þ ð c2 Þ ¼ 0:57 AE 0:23. The dependence of R c on the Feynman-x of the J=c , x J=c F , and its transverse momentum, p J=c T , is studied, as well as its dependence on the atomic number, A, of the target. For the first time, an extensive study of possible biases on R c and R 12 due to the dependence of acceptance on the polarization states of J=c and c is performed. By varying the polarization parameter, obs , of all produced J=c 's by two sigma around the value measured by HERA-B, and considering the maximum variation due to the possible c1 and c2 polarizations, it is shown that R c could change by a factor between 1.02 and 1.21 and R 12 by a factor between 0.89 and 1.16.