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Math Help - List of ideals

  1. #1
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    List of ideals

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

    Thanks.
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  2. #2
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    Quote Originally Posted by KevinKH View Post
    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.
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  3. #3
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    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?
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  4. #4
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    Quote Originally Posted by KevinKH View Post
    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}.
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