  We compute the Type Ia supernova rates in typical elliptical galaxies by
varying the progenitor models for Type Ia supernovae. To do that a formalism
which takes into account the delay distribution function (DTD) of the explosion
times and a given star formation history is adopted. Then the chemical
evolution for ellipticals with baryonic initial masses $10^{10}$, $10^{11}$ and
$10^{12} M_{\odot}$ is computed, and the mass of Fe produced by each galaxy is
precisely estimated. We also compute the expected Fe mass ejected by
ellipticals in typical galaxy clusters (e.g. Coma and Virgo), under different
assumptions about Type Ia SN progenitors. As a last step, we compute the cosmic
Type Ia SN rate in an unitary volume of the Universe by adopting several cosmic
star formation rates and compare it with the available and recent observational
data. Unfortunately, no firm conclusions can be derived only from the cosmic
SNIa rate, neither on SNIa progenitors nor on the cosmic star formation rate.
Finally, by analysing all our results together, and by taking into account
previous chemical evolution results, we try to constrain the best Type Ia
progenitor model. We conclude that the best progenitor models for Type Ia SNe
are still the single degenerate model, the double degenerate wide model, and
the empirical bimodal model. All these models require the existence of prompt
Type Ia supernovae, exploding in the first 100 Myr since the beginning of star
formation, although their fraction should not exceed 15-20% in order to fit
chemical abundances in galaxies.
