Finding all maximal ideals

Question

Let $R=\mathbb{C}[x,y]$ and $I=(x^2+y^2-1,x+y-1)$.

Find all the maximal ideals of $R/I$??

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i don't understand how evaluate all maximal ideals????

Answer 1

Let me abuse notation and use $x,y$ to denote the images of $x,y$ in the quotient $A:=R/I$. Then in $R/I$, you have two equations $y=1-x$ and $x^2+y^2=1$, so you can use those to show $x^2=x$. So $x,y$ are both idempotents and $xy=0$! These are called orthogonal idempotents.

On the other hand, clearly $x,y$ generate $A$ as a $\mathbb{C}$-algebra. Think about what happens when idempotents generate a ring: for example the ideal $Ax$ is a subring, and in fact it's a unital ring with $x$ as its identity element. Similarly $Ay$ is a subring with identity $y$, and $A=Ax\oplus Ay$.

So what are the rings $Ax$ and $Ay$? What are their ideals? How do the ideals of $Ax$ and $Ay$ combine to give (all the) ideals of $A$?

(Hint: I'm pretty sure $Ax\simeq \mathbb{C}\simeq Ay$. Therefore, if you understand the above argument, you can get $A\simeq \mathbb{C}\oplus \mathbb{C}$. What are the ideals of $A$? The geometric interpretation is that $A$ is the function ring on the intersection of two shapes $x^2+y^2=1$ and $x+y=1$ --- this is a line and a circle, so it makes sense that you only get two points, which is why $A$ splits up into a sum of two $1$-dimensional things.)