  The nature of the Type Ia supernovae (SNIa) progenitors remains still
uncertain. This is a major issue for galaxy evolution models since both
chemical and energetic feedback play a major role in the gas dynamics, star
formation and therefore in the overall stellar evolution. The progenitor models
for the SNIa available in the literature propose different distributions for
regulating the explosion times of these events. These functions are known as
the Delay Time Distributions (DTDs). This work is the first one in a series of
papers aiming at studying five different DTDs for SNIa. Here, we implement and
analyse the Single Degenerate scenario (SD) in galaxies dominated by a rapid
quenching of the star formation, displaying the majority of the stars
concentrated in the bulge component. We find a good fit to both the present
observed SNIa rates in spheroidal dominated galaxies, and to the [O/Fe] ratios
shown by the bulge of the Milky Way. Additionally, the SD scenario is found to
reproduce a correlation between the specific SNIa rate and the specific star
formation rate, which closely resembles the observational trend, at variance
with previous works. Our results suggest that SNIa observations in galaxies
with very low and very high specific star formation rates can help to impose
more stringent constraints on the DTDs and therefore on SNIa progenitors.
