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.
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