The aim of this work is the computation of the cosmic Type Ia supernova rates at very high redshifts (z>2). We adopt various progenitor models in order to predict the number of explosions in different scenarios for galaxy formation and to check whether it is possible to select the best delay time distribution model, on the basis of the available observations of Type Ia supernovae. We also computed the Type Ia supernova rate in typical elliptical galaxies of different initial luminous masses and the total amount of iron produced by Type Ia supernovae in each case. It emerges that: it is not easy to select the best delay time distribution scenario from the observational data and this is because the cosmic star formation rate dominates over the distribution function of the delay times; the monolithic collapse scenario predicts an increasing trend of the SN Ia rate at high redshifts whereas the predicted rate in the hierarchical scheme drops dramatically at high redshift; for the elliptical galaxies we note that the predicted maximum of the Type Ia supernova rate depends on the initial galactic mass. The maximum occurs earlier (at about 0.3 Gyr) in the most massive ellipticals, as a consequence of downsizing in star formation. We find that different delay time distributions predict different relations between the Type Ia supernova rate per unit mass at the present time and the color of the parent galaxies and that bluer ellipticals present higher supernova Type Ia rates at the present time.