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I’m looking for some hints. Let J be an ideal of $R \times S$. I have already proved that $$J_1 = \{a \in R / (a,b) \in J \; for \; some \; b \in S\}$$ $$J_1 = \{b \in S / (a,b) \in J \; for \; some \; a \in R\}$$ are ideals of R and S respectively.

Also, I proved that if $J_1$ and $J_2$ are ideals of R and S respectively, then $J_1 \times J_2$ Is an ideal of $R \times S$. But I don’t know how to conclude the proof.

mug_donut
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1 Answers1

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I assume the rings have identity.

By construction, we have $J\subseteq J_1\times J_2$.

Conversely, if $(a,b)\in J_1\times J_2$, then there's a $b'$ with $(a,b')\in J$, it follows $b'\in J_2$, thus also $b-b'\in J_2$. It means there's an $a'$ with $(a', b-b')\in J$, but then also $(0,b-b')=(0,1)\cdot(a',b-b')\in J$.

Finally, we have $(a,b)=(a,b')+(0,b-b')\, \in J$.

Berci
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