For R being a commutative regular excellent Noetherian ring of finite Krull dimension which conditions on $f\in R$ can ensure that the ring $R/(f)$ is regular (so, I want a sufficient condition)? I do not want to look at all points of $R$.

Upd1. My $R$ is the inductive limit of a system of inclusions of regular Noetherian rings $R_i$ of finite Krull dimension. So, I want a 'finite number of conditions' on $f$ since I would like to check them for $R_i$.

Upd2. Since I am interested in algebras over fields, it seems that regularity can be characterized in terms of Andre-Quillen homology. Yet my algebras are not of finite type; are there any 'finite' substitute for these homology groups that ensure regularity (I do not need a necessary and sufficient criterion).