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Finitely generated ideals

  1. Feb 20, 2009 #1
    Hello,

    Let [a,b] be the ideal generated by a and b.

    If R is a commutative ring with unity, let a be in R and m,n be natural numbers.

    Show that [ (a^m)-1, (a^n)-1 ] = [ (a^gcd(m,n)) -1 ]


    Seems simple but I am having trouble with it. Thanks in advance!
     
  2. jcsd
  3. Feb 23, 2009 #2
    LHS included in RHS:
    Let d=gcd(m,n), m=ud, n=vd. Apply the formula
    (x-1)(x^(k-1)+x^(k-2)+...+x+1)=x^k-1
    to x=a^d, k=u,v.

    RHS included in LHS:
    Let d=gcd(m,n)=rm-sn with positive r,s and A=a^m, B=a^n
    (a^d)-1 = (A^r)-1 - (a^d)((B^s)-1) =
    = (A^(r-1)+...+A+1)((a^m)-1) - (a^d)(B^(s-1)+...+B+1)((a^n)-1)
     
    Last edited: Feb 24, 2009
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