It feels like the answer should be obvious. It seems clear that $(x+1,y+1)\neq\mathbb{C}[x,y]$ should be the case, but I am having a surprising amount of difficulty showing it rigorously.
2026-03-29 04:41:31.1774759291
Is $(x+1, y+1)$ a proper ideal of $\mathbb{C}[x,y]$?
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Here's a cute proof. Suppose that it's not proper, i.e. there are polynomials $f$ and $g$ such that $$(x+1)f(x,y)+(y+1)g(x,y)=1$$ Substitute $x=y$ to obtain: $$(x+1)(f(x,x)+g(x,x))=1$$ but the left hand side is either $0$ or has at least degree 1.