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There is a question which states that $\operatorname{Spec}(\mathbb{C}[x,y]/(xy))$ consists of three prime ideals: $0, (x), (y)$

I want to find irreducible and connected components of $\operatorname{Spec}(\mathbb{C}[x,y]/(xy))$.

Here is another question, without an answer till now, concerning irreducible components of $\operatorname{Spec}\mathbb{C}[x]$.

Looks like the answer here is: $(x), (y)$ are irreducible components and $(x)\cup (y)$ is the connectivity component. I have inferred it from the picture of a cross on a plane, where $(x)$ coincides with horizontal line and $(y)$ with the vertical one.

Could you give me a recipe of finding the components in this case?

UPD: I have an idea: since I know the definition of Zariski closed set, then I can write down the inclusions explicitly and check the components by hand.

user26857
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Lada Dudnikova
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    In general you would take the ideal $(xy)$ and factor it into primary components. Then look at the radicals of each of the factors. In your case $(xy)=(x)\cap(y)$ is the primary decomposition, and the factors $(x)$ and $(y)$ are already prime. – user647486 Apr 17 '19 at 17:27
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    Horizontal and vertical means whatever you want, but to clarify $V(x)$ would be the $Y$-axis in Cartesian coordinates. This is, the set of points where $x=0$. Likewise, $V(y)$ would be the $X$-axis. – user647486 Apr 17 '19 at 17:33

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I see that $\mathrm{Spec}$ is connected. And I see that $(x)$, $(y)$ can not be represented as a union of two sets, so they are irreducible, and $(xy)$ is reducible.

user26857
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Lada Dudnikova
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