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Math Help - comaximal(coprime)

  1. #1
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    comaximal(coprime)

    Let A be a commutative ring with unity.
    If M,N are distinct maximal ideals of A, then
    (1) M+N=A.
    (2) M^a+N^b=A (a,b\ge1).
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  2. #2
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    Quote Originally Posted by KaKa View Post
    Let A be a commutative ring with unity.
    If M,N are distinct maximal ideals of A, then
    (1) M+N=A.
    (2) M^a+N^b=A (a,b\ge1).
    For (1), the sum of ideals is again an ideal (link). Thus, M+N is an ideal containing M. By hypthesis, M+N should properly contain an maximal ideal M. Thus M+N=A.

    For (2), every proper ideal in A is contained in a maximal ideal in A and note that A contains the unity (link).
    Assume M^a+N^b , a,b\ge1 is a proper ideal in A. Then, M^a+N^b ,a,b\ge1 should be contained in a maximal ideal. It follows that M^a+N^b , a,b\ge1 should be contained in either M or N (check their intersection). Contradiction !
    Thus, M^a+N^b , a,b\ge1 is an ideal in A which is not a proper ideal in A. We conclude that M^a+N^b=A ,a,b\ge1.
    Last edited by aliceinwonderland; March 16th 2010 at 10:48 AM.
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