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  1. #1
    Senior Member Sampras's Avatar
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    classes

    Say you want to show that  (3) \subseteq I  \subseteq \mathbb{Z} then  I = (3) or  I = \mathbb{Z} . Well (3) is principal. So it is maximal. Thus the result follows? Are there any other ways of showing this?
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  2. #2
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    Quote Originally Posted by Sampras View Post
    Say you want to show that  (3) \subseteq I \subseteq \mathbb{Z} then  I = (3) or  I = \mathbb{Z} . Well (3) is principal. So it is maximal. Thus the result follows? Are there any other ways of showing this?
    Supose (3) < I ==> there's a non-multiple of 3 in I, say x ==> since 3 is prime, (x,3) = 1 ==> there exist integers m, n s.t. 3m + xn = 1.
    But 3m in (3) < I and xn in I ==> 1 = 3m + xn in I ==> I = Z and we're done.

    Tonio
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  3. #3
    Senior Member Sampras's Avatar
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    Is (3) a coset?
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  4. #4
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    Quote Originally Posted by Sampras View Post
    Is (3) a coset?

    (3) is exactly what you said it is: the principal ideal generated by 3!!

    Tonio
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  5. #5
    Senior Member Sampras's Avatar
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    Then why are you saying: "(3) < Z?" Why not (3) subseteq Z.
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  6. #6
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    Quote Originally Posted by Sampras View Post
    Then why are you saying: "(3) < Z?" Why not (3) subseteq Z.

    Errr...because that's the standard notation for ideals of rings and or subgroups of groups: I is an ideal of ring R can be written I < R...

    Tonio
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