Find all ideals of $K \times K$ where K is a field

Question

I know that there is no nontrivial ideals of any field $K$. Again $K \times 0$ and $0 \times K$ are nontrivial ideals of $K \times K$. But how can we give a complete classification for all ideals of $K \times K$? Thanks!

Answer 1

Hint: If an ideal contains the element $(a,b)$ where $a \ne 0$ and $b\ne 0$, what else must it contain?

Answer 2

If $A_1$ and $A_2$ are rings, every ideal $I$ in $A_1 \times A_2$ can be written as $I_1 \times I_2$; now a field $\mathbb{K}$ has only two ideals, $(0)$ and $\mathbb{K}$. Endstory.

Answer 3

The image of an ideal under the canonical projection homomorphisms onto each factor would also be an ideal by definition of a homomorphism and by the first isomorphism theorem, all ideals arise in this way, so looking at

$$\pi_1:\begin{cases}K\times K\to K \\ (a,b)\mapsto a\end{cases}$$ $$\pi_2:\begin{cases}K\times K\to K \\ (a,b)\mapsto b\end{cases}$$

and noting that the only ideals of $K$ are $\{0\}$ and $K$, you get the ideals to be

$$\{\pi_1^{-1}(I)\times\pi_2^{-1}(J)\}=\{\{0\}\times\{0\}, \{0\}\times K, K\times\{0\}, K\times K\}$$

where here, $I,J\in\{\{0\},K\}$.