I made this ring and I need the text to wrap around it. I tried the shrinkwrap modifer but the text does not go around the ring entirely. Here is a picture to help illustrate the issue.
Though for this, I’d recommend a bump/texturing solution, if you’re recreating ‘the one ring’ with text as the image proposes. I don’t really see a good reason to use geometry for this… 
How would I make the material into a UV map? And then add the writing as a texture
If the ring is a manipulated torus, you already got the base UV mapping and it’s all about texture placement. How to apply the texture is dependent on if you render using Blender Internal, ‘BI’, or the new Cycles renderer. Here’s a tutorial about texturing a barrel that might be helpful if your using BI. http://cgcookie.com/blender/2009/06/22/creating-a-wooden-barrel/