We have used the Submillimeter Array to image a flux limited sample of seven submillimeter galaxies, selected by the AzTEC camera on the JCMT at 1.1 mm, in the COSMOS field at 890 µm with ∼ 2 ′′ resolution. All of the sources -two radio-bright and five radio-dim -are detected as single point-sources at high significance (> 6σ), with positions accurate to ∼ 0.2 ′′ that enable counterpart identification at other wavelengths observed with similarly high angular resolution. All seven have IRAC counterparts, but only two have secure counterparts in deep HST/ACS imaging. As compared to the two radio-bright sources in the sample, and those in previous studies, the five radio-dim sources in the sample (1) have systematically higher submillimeter-to-radio flux ratios, (2) have lower IRAC 3.6-8.0 µm fluxes, and (3) are not detected at 24µm. These properties, combined with size constraints at 890 µm (θ ∼ < 1.2 ′′ ), suggest that the radio-dim submillimeter galaxies represent a population of very dusty starbursts, with physical scales similar to local ultraluminous infrared galaxies, and an average redshift higher than radio-bright sources.