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Consider the unitary commutative ring R := K x K where K is a field. The sum and product are defined as $$(a,b) + (c,d) = (a + c, b + d)$$ and $$(a,b) . (c,d) = (ac, bd)$$

How do you find all ideals of this ring?

For a field K, I know that if I < K and I is an ideal of K then I = 0 or I = K. Does it suffice to show that for a,b,c,d $\in$ K, (a,b), (c,d) $\in$ R, since K is closed so (ac, bd) $\in$ R ? So R is a field, or do I need to show every (ac, bd) is a unit? so it's only ideals are itself and 0?

Shmulley Boteach
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  • $R$ has zero divisors so it has additional ideals. Work out what the zero divisors are and you'll probably be able to identify the non-trivial ideals. – Robert Shore Feb 22 '21 at 10:47
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    In $K$ you have two possibilities for an element $a$: it is invertible or $0$. In $K\times K$, an element $(a,b)$ has then four possibilities: $a=0$ and $b$ invertible, $a=0$ and $b=0$, etc. Try to see what that means for ideals. – Captain Lama Feb 22 '21 at 10:49

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Hint: $K \times K \cong K[x]/(x^2-x)$

lhf
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