I'm trying to understand an example in my lecture notes, it states that $Spec(\mathbb{R}[X,Y]/(X-Y^2,X+Y))$ has only two points. Can anyone develop more please ? and how would these points be like ?
Thanks for the help.
2026-03-25 20:15:48.1774469748
prime ideals of $\mathbb{R}[X,Y]/(X-Y^2,X+Y)$
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The maximal ideals containing $(X-Y^2,X+Y)$ are $(X,Y)$ and $(X-1,Y+1)$. To see this, the ideal $(X-Y^2,X+Y)$ has $Y^2+Y$ has an element, and so either $Y$ or $Y+1$. If it contains $Y$ it contains $X$, and if it contains $Y+1$, it contains $X-1$.