  We study the metallicities and abundance ratios of early-type galaxies in
cosmological semi-analytic models (SAMs) within the hierarchical galaxy
formation paradigm. To achieve this we implemented a detailed galactic chemical
evolution (GCE) model and can now predict abundances of individual elements for
the galaxies in the semi-analytic simulations. This is the first time a SAM
with feedback from Active Galactic Nuclei (AGN) has included a chemical
evolution prescription that relaxes the instantaneous recycling approximation.
We find that the new models are able to reproduce the observed mass-metallicity
(M*-[Z/H]) relation and, for the first time in a SAM, we reproduce the observed
positive slope of the mass-abundance ratio (M*-[$alpha$/Fe]) relation. Our
results indicate that in order to simultaneously match these observations of
early-type galaxies, the use of both a very mildly top-heavy IMF (i.e., with a
slope of x=1.15 as opposed to a standard x=1.3), and a lower fraction of
binaries that explode as Type Ia supernovae appears to be required. We also
examine the rate of supernova explosions in the simulated galaxies. In
early-type (non-star forming) galaxies, our predictions are also consistent
with the observed SNe rates. However, in star-forming galaxies, a higher
fraction of SN Ia binaries than in our preferred model is required to match the
data. If, however, we deviate from the classical model and introduce a
population of SNe Ia with very short delay times, our models simultaneously
produce a good match to the observed metallicities, abundance ratios and SN
rates.
