The extension in the ring will be $\frac{\mathrm{mgR}}{\mathrm{aY}}$

The tension in the ring will be same throughout.

Elastic potential energy stored in the ring will be $\frac{{\mathrm{m}}^2{\mathrm{g}}^2\mathrm{R}}{8\mathrm{\pi Ya}}$

The tension in the ring will be independent of the radius of ring if radius of the ring is less than 2R.