ring-around-a-rosy vs ring-around-the-rosy

ring-around-a-rosy and ring-around-the-rosy both are nouns.

Word	Noun	Adjective	Verb	Adverb
ring-around-a-rosy	Yes	No	No	No
ring-around-the-rosy	Yes	No	No	No

As nouns, ring-around-a-rosy and ring-around-the-rosy are synonyms defined as:

ring-around-a-rosy and ring-around-the-rosy: a children's game in which the players dance around in a circle and at a given signal all squat

Other synonyms of ring-around-a-rosy include ring-a-rosy.

ring-around-a-rosy (noun)	ring-around-the-rosy (noun)
a children's game in which the players dance around in a circle and at a given signal all squat	a children's game in which the players dance around in a circle and at a given signal all squat