We study the stellar populations of SNe Ia host galaxies using SDSS-II spectroscopy. We focus on the relationships of SNe Ia properties with stellar velocity dispersion and the stellar population parameters age, metallicity and element abundance ratios derived by fitting absorption line indices to stellar population models. We concentrate on a sub-sample of 84 SNe Ia from the SDSS-II Supernova Survey. In agreement with previous findings, we find that SALT2 stretch factor values show the strongest dependence on stellar population age. Hence, SNe Ia peak-luminosity is closely related to the age of the stellar progenitor systems, where more luminous SNe Ia appear in younger stellar populations. We find no statistically significant trends in the Hubble residual with any of the stellar population parameters studied, including age and metallicity contrary to the literature, as well as with stellar velocity dispersion. Moreover, we find that the method of stellar mass derivation is affecting the Hubble residual-mass relationship when lower number statistics are used. We extend the sample to also include SNe Ia with available SDSS host galaxy photometry only. For this larger sample (247 objects) the reported Hubble residual-mass relation is strongly dependent on the stellar mass range studied and behaves as a step function. In the high mass regime, probed by our host spectroscopy sample, the relation between Hubble residual and stellar mass is flat. Below a stellar mass of ~2x10^10 Msun, i.e. close to the evolutionary transition mass of low-redshift galaxies reported in the literature, the trend changes dramatically such that lower mass galaxies possess lower luminosity SNe Ia after light-curve corrections. This non-linear behaviour of the Hubble residual-mass relationship should be accounted for when using stellar mass as a further parameter for minimising the Hubble residuals.
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