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I am wondering how to prove an ideal of a ring $R$ which is defined as a coordinates. Let $R$ be the ring of $\mathbb{Z} \times \mathbb{Z}$. Let $I={(a,a)| a\in \mathbb{Z}}$. I determine that the $I$ is a subring of $R$. Next step is to show the multiplication between the elements of $R$ and $I$. But I have read in the book that I need worried about the elements of $R$ and not just $I$.

Thanks,

Cbarker1