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We compute the Type Ia supernova rates in typical elliptical galaxies byvarying the progenitor models for Type Ia supernovae. To do that a formalismwhich takes into account the delay distribution function (DTD) of the explosiontimes and a given star formation history is adopted. Then the chemicalevolution 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 isprecisely estimated. We also compute the expected Fe mass ejected byellipticals in typical galaxy clusters (e.g. Coma and Virgo), under differentassumptions about Type Ia SN progenitors. As a last step, we compute the cosmicType Ia SN rate in an unitary volume of the Universe by adopting several cosmicstar formation rates and compare it with the available and recent observationaldata. Unfortunately, no firm conclusions can be derived only from the cosmicSNIa rate, neither on SNIa progenitors nor on the cosmic star formation rate.Finally, by analysing all our results together, and by taking into accountprevious chemical evolution results, we try to constrain the best Type Iaprogenitor model. We conclude that the best progenitor models for Type Ia SNeare still the single degenerate model, the double degenerate wide model, andthe empirical bimodal model. All these models require the existence of promptType Ia supernovae, exploding in the first 100 Myr since the beginning of starformation, although their fraction should not exceed 15-20% in order to fitchemical abundances in galaxies.

Galactic and Cosmic Type Ia SN rates: is it possible to impose constraints on SNIa progenitors?