Using the sample of Type Ia supernovae (SNe Ia) discovered by the Hubble Space Telescope (HST) Cluster Supernova Survey and augmented with HST-observed SNe Ia in the GOODS fields, we search for correlations between the properties of SNe and their host galaxies at high redshift. We use galaxy color and quantitative morphology to determine the red sequence in 25 clusters and develop a model to distinguish passively evolving early-type galaxies from star-forming galaxies in both clusters and the field. With this approach, we identify six SN Ia hosts that are early-type cluster members and eleven SN Ia hosts that are early-type field galaxies. We confirm for the first time at z>0.9 that SNe Ia hosted by early-type galaxies brighten and fade more quickly than SNe Ia hosted by late-type galaxies. We also show that the two samples of hosts produce SNe Ia with similar color distributions. The relatively simple spectral energy distributions (SEDs) expected for passive galaxies enable us to measure stellar masses of early-type SN hosts. In combination with stellar mass estimates of late-type GOODS SN hosts from Thomson & Chary (2011), we investigate the correlation of host mass with Hubble residual observed at lower redshifts. Although the sample is small and the uncertainties are large, a hint of this relation is found at z>0.9. By simultaneously fitting the average cluster galaxy formation history and dust content to the red-sequence scatters, we show that the reddening of early-type cluster SN hosts is likely E(B-V) <~ 0.06. The similarity of the field and cluster early-type host samples suggests that field early-type galaxies that lie on the red sequence may also be minimally affected by dust. Hence, the early-type hosted SNe Ia studied here occupy a more favorable environment to use as well-characterized high-redshift standard candles than other SNe Ia.
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