Let $R$ be a noetherian regular domain. Suppose that $a, b \in R$, with $b \neq 0$, and consider the ring $S:=R[\frac{a}{b}]=R[X]/(bXa)$. Is $S$ regular? If this is not the case are there some conditions on $a$ and $b$ (or on $R$) that imply regularity? For example $a=1$ is enough.
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If $R = k[x,y]$ and we throw in $\frac {x^2}y$, we get $k[x,y,z]/(zyx^2)$, not regular. This shows that it's not enough to assume $a,b$ form a regular sequence. The only sufficient condition I can come up with is that $a$ be outside the square of (every/the) maximal ideal 

