ring (n.1) Look up ring at Dictionary.com
"circular band," Old English hring "small circlet, especially one of metal for wearing on the finger or as part of a mail coat; anything circular," from Proto-Germanic *hringaz "something curved, circle" (cognates: Old Norse hringr, Old Frisian hring, Danish, Swedish, Dutch ring, Old High German hring, German Ring), from PIE *(s)kregh- nasalized form of (s)kregh-, from root *(s)ker- (3) "to turn, bend," with wide-ranging derivative senses (cognates: Latin curvus "bent, curved," crispus "curly;" Old Church Slavonic kragu "circle," and perhaps Greek kirkos "ring," koronos "curved").

Other Old English senses were "circular group of persons," also "horizon." Meaning "place for prize fight and wrestling bouts" (early 14c.) is from the space in a circle of bystanders in the midst of which such contests once were held, "... a circle formed for boxers, wrestlers, and cudgel players, by a man styled Vinegar; who, with his hat before his eyes, goes round the circle, striking at random with his whip to prevent the populace from crowding in" [Grose, 1785]. Meaning "combination of interested persons" is from 1829. Of trees, from 1670s; fairy ring is from 1620s. Ring finger is Old English hringfingr, a compound found in other Germanic languages. To run rings round (someone) "be superior to" is from 1891.

Nursery rhyme ring a ring a rosie is attested in an American form (with a different ending) from c.1790. "The belief that the rhyme originated with the Great Plague is now almost universal, but has no evidence to support it and is almost certainly nonsense" ["Oxford Dictionary of English Folklore"]. This proposal of connection dates only to the late 1960s.
ring (v.1) Look up ring at Dictionary.com
"sound a bell," Old English hringan "sound, give a certain resonant sound when struck; announce by bells," from Proto-Germanic *khrengan (cognates: Old Norse hringja, Swedish ringa, Middle Dutch ringen), probably of imitative origin. Related: Rang; rung. Originally a weak verb, strong inflection began in early Middle English by influence of sing, etc. To ring down a theatrical curtain is from 1772, from the custom of signaling for it by ringing a bell. To ring up a purchase on a cash register is by 1937, from the bell that sounded. Specialized sense "give a resonant sound when struck as an indication of genuineness or purity," with transferred use (as in to ring hollow) is from 1610s.
ring (v.2) Look up ring at Dictionary.com
"make a circle around," Old English ymbhringan, from the root of ring (n.1). Intransitive sense "gather in a ring" is mid-15c. Sense of "provide or attach a ring" is late 14c. Meaning "move in a circle around" is from 1825. Related: Ringed; ringing. Compare Frisian ringje, Middle Dutch and Dutch ringen, Old High German ringan, German ringen, Old Norse hringa, hringja.
ring (n.2) Look up ring at Dictionary.com
1540s, "set of church bells," from ring (v.1). Meaning "a call on the telephone" is from 1900; to give (someone) a ring "call on the telephone" was in use by 1910. Meaning "a ringing tone" is from 1620s; specifically "the ringing sound made by a telephone" by 1951. Meaning "resonance of coin or glass as a test of genuineness" is from 1850, with transferred use (ring of truth, etc.).