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 [When is a blow-up non-singular?](https://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."](http://www.informaworld.com/smpp/content~db=all~content=a780044502 "On the smoothness of blow-ups") 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. 



If the quotient of a local ring is regular, does that imply that the original ring must be regular?
The quoted result relies on the following elementary characterization of local regular rings:



> 
> Let $R$ be a local ring with maximal ideal $\mathfrak m$ and $x\in\mathfrak m-\mathfrak m^2$. Then $R$ is regular iff $R/(x)$ is regular and $x$ doesn't belong to any minimal prime.
> 
> 
> 


