  In our quest to identify the progenitors of Type Ia supernovae (SNe Ia), we
first update the nucleosynthesis yields both for near-Chandrasekhar (Ch) and
sub-Ch mass white dwarfs (WDs), for a wide range of metallicity, with our
two-dimensional hydrodynamical code and the latest nuclear reaction rates. We
then include the yields in our galactic chemical evolution code to predict the
evolution of elemental abundances in the solar neighborhood and dwarf
spheroidal (dSph) galaxies: Fornax, Sculptor, Sextans, and Carina. In the
observations of the solar neighborhood stars, Mn shows an opposite trend to
$\alpha$ elements, showing an increase toward higher metallicities, which is
very well reproduced by deflagration-detonation transition of Ch-mass WDs, but
never by double detonations of sub-Ch-mass WDs alone. The problem of Ch-mass
SNe Ia was the Ni over-production at high metallicities. However, we found that
Ni yields of Ch-mass SNe Ia are much lower with the solar-scaled initial
composition than in previous works, which keeps the predicted Ni abundance
within the observational scatter. From the evolutionary trends of elemental
abundances in the solar neighborhood, we conclude that the contribution of
sub-Ch-mass SNe Ia in chemical enrichment is up to 25\%. In dSph galaxies,
however, larger enrichment from sub-Ch-mass SNe Ia than in the solar
neighborhood may be required, which causes a decrease in [(Mg, Cr, Mn, Ni)/Fe]
at lower metallicities. The observed high [Mn/Fe] ratios in Sculptor and Carina
may also require additional enrichment from pure deflagrations, possibly as
Type Iax supernovae. Future observations of dSph stars will provide more
stringent constraints on the progenitor systems and explosion mechanism of SNe
Ia.
