I'm trying to show the following claim:
Let $R$ be a Noetherian ring. $R$ satisfies Serre's condition $S_k$ if and only if whenever $I$ is an ideal of the principal class (i.e., $I$ can be generated by $\operatorname{ht}I$ elements) of height at most $ k-1$, then $I$ is unmixed in the sense that every associated prime of $R/I$ is minimal over $I$.
I prove the result when $I=0$ and $k=1$ which is a standard result and I know the result holds for $S_2$, but don't know how to translate the ideal of principal class condition: I took an associated prime of $R/I$ localized at it and noticed that depth($R/I)_\mathfrak{p}=0$. If I can somehow translate the condition $S_k$ to $R/I$ then $I$ would be done by Cohen-Macaulayness. Any ideas, suggestions or references? Thanks!