  Type Ia Supernovae (SNe Ia) are widely used to measure the expansion of the
Universe. To perform such measurements the luminosity and cosmological redshift
($z$) of the SNe Ia have to be determined. The uncertainty on $z$ includes an
unknown peculiar velocity, which can be very large for SNe Ia in the virialized
cores of massive clusters. We determine which SNe Ia exploded in galaxy
clusters. We then study how the correction for peculiar velocities of host
galaxies inside the clusters improves the Hubble residuals. Using 145 SNe Ia
from the Nearby Supernova Factory we found 11 candidates for membership in
clusters. To estimate the redshift of a cluster we applied the bi-weight
technique. Then, we use the galaxy cluster redshift instead of the host galaxy
redshift to construct the Hubble diagram. For SNe Ia inside galaxy clusters the
dispersion around the Hubble diagram when peculiar velocities are taken into
account is smaller in comparison with a case without peculiar velocity
correction, with a $wRMS=0.130\pm0.038$ mag instead of $wRMS=0.137\pm0.036$
mag. The significance of this improvement is 3.58 $\sigma$. If we remove the
very nearby Virgo cluster member SN2006X ($z<0.01$) from the analysis, the
significance decreases to 1.34 $\sigma$. The peculiar velocity correction is
found to be highest for the SNe Ia hosted by blue spiral galaxies, with high
local specific star formation rate and smaller stellar mass, seemingly counter
to what might be expected given the heavy concentration of old, massive
elliptical galaxies in clusters. As expected, the Hubble residuals of SNe Ia
associated with massive galaxy clusters improve when the cluster redshift is
taken as the cosmological redshift of the SN. This fact has to be taken into
account in future cosmological analyses in order to achieve higher accuracy for
cosmological redshift measurements. Here we provide an approach to do so.
