Can you be more specific  on the meaning of affine ring? More precisely do you mean the ring of function on an *arbitrary*  affine variety or just the ring of functions on $\mathbb{C}^n$? In any case,  this [nice  article of R. Stanley][1] may be a good place to start.


  [1]: http://www.ams.org/journals/bull/1979-01-03/S0273-0979-1979-14597-X/home.html