Let $R$ be a Noetherian domain of finite Krull dimension. Let $0\ne g \in R$ be such that $R/gR$ is reduced. let $q$ be a positive integer.
Is the following true: $R[X]/(X^q - g)$ is a regular ring if and only if $R$ and $R/gR$ are regular ?
Note: A ring is called regular iff localisation at every prime ideal is regular local ring.
If needed, I'm willing to assume $R$ contains an algebraically closed field.