The scratch on the cornea caused an ulceration. In order to heal, blood vessels must grow into the area, as the cornea has no blood supply. A month of antibiotics should have removed all susceptible bacteria. The white ring is probably the remains of the vascularization, which should mostly disappear when the eye is completely healed. If there is still redness in the eye, give homeopathic Euphrasia twice a day until healed. If there is still swelling, use homeopathic Apis as well.