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.
Prove polynomial ring over a discrete valuation ring quotient by powers of maximal ideal is regular?
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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 '19 at 15:50
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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 '19 at 15:55
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1@user26857 Thank you very much, it's very helpful. – Z Wu Feb 17 '19 at 15:58