How to prove an ideal of a ring R which is defined as a coordinates

[cbarker1]

Hi Everyone,

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

 

[GJA]

Hi Cbarker1,

Assuming that multiplication in $R$ is defined as $(a_{1},a_{2})\cdot (b_{1},b_{2}) = (a_{1}b_{1},a_{2}b_{2}),$ $I$ is not an ideal of $R$. For example $(1,0)\cdot(a,a) = (a,0)\notin I$ for any $a\neq 0$.