Suppose $R$ is a local ring and let $I\subset R$ be some nontrivial ideal. Are there conditions that we can place on $I$ so that if $R/I$ is regular, then so is $R$?  

I am aware of the result that states: if $R$ is already a regular local ring, then $R/I$ is regular iff $I$ is generated by a subset of a regular system of parameters.  But I am wondering more about whether or not the regularity of $R$ itself can be determined by it's quotient $R/I$.  

Background for this question: after reading the responses to the question http://mathoverflow.net/questions/21232/when-is-a-blow-up-non-singular I am trying to work through the first argument in the second section of the paper ["On the smoothness of blow-ups."][1] I think the author uses the result that I am asking about when they state: "$S_P/f_i S_P$ is a regular local ring; since $f_i$ is a non-zero-divisor on $S_P$, it follows that $S_P$ is itself regular."  I am just trying to figure out where this result comes from.  

  [1]: http://www.informaworld.com/smpp/content~db=all~content=a780044502 "On the smoothness of blow-ups"