Let y be an nth root of 1 and adjoin y to R. Let's assume y is not already in R, else there isn't much to talk about. We can adjoin y to R, or we can adjoin y to Z and then bring in the rest of R. either way the result is R[y]. Thus it is enough to understand the cyclotomic extensions of Z.
A similar result holds when R has characteristic m. Adjoin y to Zm and produce a subring of R[y]. Without loss of generality, we will consider the cyclotomic extensions of the integers, or the integers mod m.
If m = ab, where a and b are relatively prime, then the ring Zm is the direct product of Za and Zb. An nth root in the direct product implies an nth root in each component, (reduce mod a or b), and nth roots in the two components can be combined to produce an nth root in Zm. therefore we can restrict attention to prime powers.
Subsequent sections will consider Zp, because that's a field. It is also the homomorphick image of Zpk, so we may be able to work our way back to prime powers, and then back to Zm.
Note that the extension is always integral, thanks to the polynomial xn-1. Any root y will have a minimum polynomial, which divides any other polynomial with root y.