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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.

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No. Let $R=k[x,y]$ and $g=xy$. Then $R[z]/(z-xy) \cong k[x,y,z]/(z-xy)$ is regular, but $R/gR \cong k[x,y]/(xy)$ is not.

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  • $\begingroup$ I see ... possibly it is true if $(g)$ is a prime ideal in $R$ ? $\endgroup$
    – uno
    Jan 22, 2020 at 17:18
  • $\begingroup$ @uno No. Let $R=k[x,y]$ and $g=x^2+y^3$. Then $R[z]/(z-x^2-y^3) \cong k[x,y,z]/(z-x^2-y^3)$ is regular, but $R/gR \cong k[x,y]/(x^2+y^3)$ is not. $\endgroup$
    – metalspringpro
    Jan 22, 2020 at 17:20

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