is expected to be dominated by compact binaries at redshifts inaccessible to direct searches for gravitational-wave events [@CallisterEA_2016]. Additionally, a detection of the  would enable a model-independent test of general relativity by discerning the polarization of gravitational waves [@TestingGR_stoch; @stoch_nongr_O1]. Because general relativity predicts only two tensor polarizations for gravitational waves, any detection of alternative polarizations would imply a modification to our current understanding of gravity [@Eardley:1973br; @Eardley:1974nw; @Will:2014kxa]. For recent reviews on relevant data analysis methods, see [@RomanoCornish; @Christensen_2018]. In this manuscript, we present a search for an isotropic stochastic background using data from Advanced LIGO’s second observing run (O2). As in previous LIGO and Virgo analyses, this search is based on cross-correlating the strain data between pairs of gravitational-wave detectors [@stoch_O1; @S5StochIso]. We first review the stochastic search methodology, then describe the data and data quality cuts. As we do not find evidence for the , we place upper limits on the possible amplitude of an isotropic , as well as limits on the presence of alternative gravitational-wave polarizations. Upper limits on anisotropic stochastic backgrounds are given in a publication that is a companion to this one [@PhysRevD.100.062001]. We then give updated forecasts of the sensitivities of future 