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Math Help - algebra - finding ideals

  1. #1
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    algebra - finding ideals

    Just wondering if anyone knows of a quick method for finding ideals from a multiplication table?

    e.g say you are looking at Z/6Z, which has ideals {0,2,4} and {0,3} is there any way you can look at the table and just spot immediately what they are?

    If anyone can share their wisdom it would be appreciated!

    Thanks in advance!
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  2. #2
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    Hello,

    If you're talking about groups with + not *, I think I can help...

    The cardinal of the ideal divides the cardinal of the original set, here it's Z/6Z
    Hence, the ideals have possible cardinals : 1,2,3,6.

    This also helps in finding the elements of the ideals : starting from 0, the following element of the ideal will be \frac{n}{\text{cardinal of the ideal}}, if we work in Z/nZ.

    For example, the ideal of Z/6Z which has 3 elements will be the set of elements x such as :
    \{x=\frac 63 k \ \big/ \ k \in \{0;1;2\} \}

    This means that the ideal of Z/6Z with 3 elements is : \{0;2;4\}



    I hope this is clear enough
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  3. #3
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    Quote Originally Posted by hunkydory19 View Post
    Just wondering if anyone knows of a quick method for finding ideals from a multiplication table?

    e.g say you are looking at Z/6Z, which has ideals {0,2,4} and {0,3} is there any way you can look at the table and just spot immediately what they are?
    When dealing with \mathbb{Z}_n all subgroups (of the addition group) are ideals (not true in general but for modulo rings it works). Futhermore, all subgroups of \mathbb{Z}_n have the form k\mathbb{Z}_n = \{ kx| x\in \mathbb{Z}_n\} where k is a divisor of n.

    Thus, in \mathbb{Z}_6, your example, the subgroups (which will turn to be ideals are:
    1\mathbb{Z}_6 = \mathbb{Z}_6
    2\mathbb{Z}_6 = \{ [0],[2],[4]\}
    3\mathbb{Z}_6 = \{ [0],[3]\}
    6\mathbb{Z}_6 =\{ [0]\}.
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