# List of ideals

Printable View

• Oct 26th 2006, 10:13 AM
KevinKH
List of ideals
Could someone help me find the complete list of ideals for the ring Z x Z?

Thanks.
• Oct 26th 2006, 10:39 AM
ThePerfectHacker
Quote:

Originally Posted by KevinKH
Could someone help me find the complete list of ideals for the ring Z x Z?

Since, $\mathbb{Z}$ is cyclic all subgroups must have form $n\mathbb{Z}$.
We note that $n\mathbb{Z}$ and $m\mathbb{Z}$ are subgroups of $\mathbb{Z}$ thus, $n\mathbb{Z} \times m\mathbb{Z}$ is an additive subgroup of $\mathbb{Z}\times \mathbb{Z}$ which is also an ideal.
• Oct 26th 2006, 11:18 AM
KevinKH
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, 05:25 PM
ThePerfectHacker
Quote:

Originally Posted by KevinKH
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?

Here are all the ideals,
$\{n\mathbb{Z}\times m\mathbb{Z}|n,m \in \mathbb{Z}\}$.
Meaning it is a set of all "linear-combinations" (abusing terminology) of $\mathbb{Z}$ and $\mathbb{Z}$.