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Ring-around-the-roses is, as mentioned in the OP, a nursery rhyme. It is also a common children's game, at least in the US. The exact words vary, as is true of all playground rhymes, but this is the version with which I'm most familiar.
1 The explanations of the rhyme's true meaning are inconsistent, and they seem to be contrived to match whichever version of Ring Around the Rosie the teller is familar with. For example, the purpose of the pocket full of posies is said to be any one of the following:  Something carried to ward off the disease.