Could someone help me find the complete list of ideals for the ring Z x Z?

Thanks.

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- Oct 26th 2006, 09:13 AMKevinKHList of ideals
Could someone help me find the complete list of ideals for the ring Z x Z?

Thanks. - Oct 26th 2006, 09:39 AMThePerfectHacker
Since, $\displaystyle \mathbb{Z}$ is cyclic all subgroups must have form $\displaystyle n\mathbb{Z}$.

We note that $\displaystyle n\mathbb{Z}$ and $\displaystyle m\mathbb{Z}$ are subgroups of $\displaystyle \mathbb{Z}$ thus, $\displaystyle n\mathbb{Z} \times m\mathbb{Z}$ is an additive subgroup of $\displaystyle \mathbb{Z}\times \mathbb{Z}$ which is also an ideal. - Oct 26th 2006, 10:18 AMKevinKH
So n and m could be written as {(0), (1), (2), (3), (4), (5),...} where (n) and (m) are {rn: r in Z} and {rm: r in Z} respectively. Is this a way of writing the complete list of ideals for Z x Z?

- Oct 26th 2006, 04:25 PMThePerfectHacker