# Prove polynomial ring over a discrete valuation ring quotient by powers of maximal ideal is regular?

Let $$(R,\mathfrak{m},k)$$ be a discrete valuation ring, (of characteristic $$p$$ if you need).

Let $$n\geq 1$$ be an integer.

Is the ring $$\frac{R}{\mathfrak{m}^n}[x]$$ regular?

Note that: Regularity can be checked at localisation of maximal ideals.

• This is part of a general phenomenon: if $A$ is regular and $a\in A$ belongs to the square of a maximal ideal $\mathfrak m$, then $A/(a)$ is not regular. – user26857 Feb 17 at 15:50
• In your case $A=R[x]$, and $a=t^n$. Then $a\in M^2$, where $M=(\mathfrak m,x)$. In fact, the localization of $A$ at $M$ is not regular. – user26857 Feb 17 at 15:55
• @user26857 Thank you very much, it's very helpful. – Lance Wu Feb 17 at 15:58