Ring is a verb, meaning to telephone or to make a sound like a bell. Adverbs come from adjectives, not verbs. The only way you can get an adverb from 'ring' is to use the present participle as an adjective, 'ringing' and add 'ly' to make the adverb 'ringingly'. The Free Dictionary says it exists, meaning 'in a ringing manner' but I have never come across this word in my whole life before. 'Ring' is also a noun, but I'm not sure how you'd make a meaningful adverb out of that.