We study the properties of Type Ia Supernovae (SNe Ia) as functions of the radial distance from their host galaxy centers. Using a sample of 62 SNe Ia with reliable luminosity, reddening, and decline rate determinations, we find no significant radial gradients of SNe Ia peak absolute magnitudes or decline rates in elliptical+S0 galaxies, suggesting that the diversity of SN properties is not related to the metallicity of their progenitors. We do find that the range in brightness and light curve width of supernovae in spiral galaxies extends to brighter, broader values. These results are interpreted as support for an age, but not metallicity, related origin of the diversity in SNe Ia. If confirmed with a larger and more accurate sample of data, the age-luminosity relation would offer a new and powerful tool to probe the ages and age gradients of stellar populations in galaxies at redshift as high as $z\sim1-2$. The absence of significant radial gradients in the peak $\rm (B-V)_0$ and $\rm (V-I)_0$ colors of SNe Ia supports the redding correction method of Phillips et al (1999). We find no radial gradient in residuals from the SN Ia luminosity-width relation, suggesting that the relation is not affected by properties of the progenitor populations and supporting the reliability of cosmological results based upon the use of SNe Ia as distance indicators.