This paper presents a study of the chemical compositions in cool gas around a sample of 27 intermediate-redshift galaxies . The sample comprises 13 massive quiescent galaxies at z = 0.40 - 0.73 probed by QSO sightlines at projected distances d = 3 - 400 kpc , and 14 star-forming galaxies at z = 0.10 - 1.24 probed by QSO sightlines at d = 8 - 163 kpc . The main goal of this study is to examine the radial profiles of the gas-phase Fe/ \alpha ratio in galaxy halos based on the observed Fe II to Mg II column density ratios . Because \mathrm { Mg ^ { + } } and \mathrm { Fe ^ { + } } share similar ionization potentials , the relative ionization correction is small in moderately ionized gas and the observed ionic abundance ratio \mathrm { \mathit { N } ( Fe II ) / \mathit { N } ( Mg II ) } places a lower limit to the underlying ( Fe/Mg ) elemental abundance ratio . For quiescent galaxies , a median and dispersion of \mathrm { log \langle \mathit { N } ( Fe II ) / \mathit { N } ( Mg II ) } \rangle _ { med } = -0 % .06 \pm 0.15 is found at d\ > ^ { < } _ { \sim } \ > 60 kpc , which declines to \mathrm { log \langle \mathit { N } ( Fe II ) / \mathit { N } ( Mg II ) } \rangle _ { med } < -0.3 at d\ > ^ { > } _ { \sim } \ > 100 kpc . On the other hand , star-forming galaxies exhibit \mathrm { log \langle \mathit { N } ( Fe II ) / \mathit { N } ( Mg II ) } \rangle = -0.25 \pm 0.21 at d\ > ^ { < } _ { \sim } \ > 60 kpc and \mathrm { log \langle \mathit { N } ( Fe II ) / \mathit { N } ( Mg II ) } \rangle = -0.9 \pm 0.4 at larger distances . Including possible differential dust depletion or ionization correction would only increase the inferred ( Fe/Mg ) ratio . The observed \mathrm { \mathit { N } ( Fe II ) / \mathit { N } ( Mg II ) } implies super-solar Fe/ \alpha ratios in the inner halo of quiescent galaxies . An enhanced Fe abundance indicates a substantial contribution by Type Ia supernovae in the chemical enrichment , which is at least comparable to what is observed in the solar neighborhood or in intracluster media but differs from young star-forming regions . In the outer halos of quiescent galaxies and in halos around star-forming galaxy , however , the observed \mathrm { \mathit { N } ( Fe II ) / \mathit { N } ( Mg II ) } is consistent with an \alpha -element enhanced enrichment pattern , suggesting a core-collapse supernovae dominated enrichment history .