15x7 is generally the narrowest rim recommended by manufacturers for a 31x10.50 tire. IMO, this is actually the best combo for offroad use, you get a better sidewall bulge that both better protects the rim's edge from impacts, and retains the bead better when running lower pressure.
A 32x11.50 will fit fine too, even though most mfgr's recommend the 8" rim for this size. I wouldn't go any wider than that though on a 7" rim.