Let $I \subseteq R$ and $J \subseteq S$ be ideals of $R$ and $S$, respectively. Then, $I \times J$ is an ideal of $R \times S$. To see this, let $(r_1, s_1), (r_2, s_2) \in I \times J$. Then, $(r_1, s_1) + (r_2, s_2) = (r_1 + r_2, s_1 + s_2) \in I \times J$ since
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