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Given two rings $R,S$ and a principal ideal $((a,b))=I\in R\times S$ where $(a,b)\in R \times S$; is it true in general that $(R\times S) / ((a,b))\cong R/(a)\times S/(b)$?

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  • $\begingroup$ There are two obvious candidates of isomorphisms in both sides. Can you show they are mutually inverse? $\endgroup$ – Mariano Suárez-Álvarez Nov 30 '11 at 7:03
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Perhaps it's more clear if you note the following: $((a,b))=(a)\times (b)$. From there you can recall the common fact that $(R\times S)/(\mathfrak{a}\times\mathfrak{b})\cong (R/\mathfrak{a})\times (S/\mathfrak{b})$.

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