A non-graded Gorenstein Artin $k$-algebra such that its associated graded ring is also Gorenstein

Question

I am wondering if there is a non-graded Gorenstein Artin $k$-algebra $R$ such that its associated graded ring, $\mathrm{gr}(R)$, is also Gorenstein.

All the non-graded Gorenstein rings I tried so far have not Gorenstein associated graded rings.

Thanks for your help.

Answer 1

Let $R=k[x,y]/(xy^2, x^3-y^2)$. Then $\text{gr}(R)=k[x,y]/(x^4,y^2)$.

$R$ is nongraded Gorenstein and $\mathrm{gr}(R)$ is also Gorenstein.