Carol, rice can be very hard on their system, so it could be difficulty with digesting the rice rather than an allergy. Lots of babies have had that problem.
As far as the allergies go, Kaelyn had eczema and weird rashes almost from the beginning. When we started solids, they got steadily worse. When she started drinking milk, it became more like hives and then finally it looked like she was covered in boils (on her stomach and legs). After the hives started , her ped did a RAST test. We got the results back just after the “boils” (not really boils, but they looked like it) started. The RAST test showed her allergies to milk and wheat. When we cut those out of her diet, she got a lot better, but the rashes were still there (and she still had a ring around her bottom). So we went to the allergist for skin testing and discovered the allergies to apples and peaches. Hope this helps!