The ring $(k[x,y]/(y^2,xy))_x$ has no nonzero nilpotents I'd like to understand why the ring $(k[x,y]/(y^2,xy))_x$ has no nonzero nilpotents. I know that since localization is exact we have $(k[x,y]/(y^2,xy))_x\cong k[x,y]_x/(y^2,xy)_x$, but I'm not sure what to do from here. What can I try?

From the right hand of the isomorphism you wrote, $(y^2,xy)_x=(y)_x$, so it looks like $k[x,y]_x/(y)_x\cong k[x]_x$, the Laurent polynomials, which is a domain.
