# List of ideals

• Oct 26th 2006, 09:13 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, 09:39 AM
ThePerfectHacker
Quote:

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

Since, $\displaystyle \mathbb{Z}$ 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 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, 04:25 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,
$\displaystyle \{n\mathbb{Z}\times m\mathbb{Z}|n,m \in \mathbb{Z}\}$.
Meaning it is a set of all "linear-combinations" (abusing terminology) of $\displaystyle \mathbb{Z}$ and $\displaystyle \mathbb{Z}$.