1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Algebra problem

  1. Sep 19, 2005 #1

    Galileo

    User Avatar
    Science Advisor
    Homework Helper

    I want to show that if I and J are coprime ideals of a ring R, so I+J=R, then for any positive numbers m and n we also have [itex]I^n+I^m=R[/itex].

    I thought the easiest way to do it was to show that [itex]1 \in I^n+J^m[/itex] given that there exist [itex]i\in I[/itex] and [itex]j\in J[/itex] such that [itex]i+j=1[/itex]. But I haven't had much luck yet. Any hint would be appreciated.
     
  2. jcsd
  3. Sep 19, 2005 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    I'm going to hope that R is commutative, for then consider (i+j) raised to some power to do with m and n but larger than both.
     
  4. Sep 19, 2005 #3

    Galileo

    User Avatar
    Science Advisor
    Homework Helper

    Yes, R is commutative. I forgot to mention that.
    I already had tried expanding (i+j) to some power. For example m+n or mn:
    [tex]1=(i+j)^{m+n}=(i+j)^m(i+j)^n=\sum_{k=0}^m {m \choose k}i^{m-k}j^k\sum_{k=0}^{n}{n \choose k}i^{n-k}j^k=\sum_{k=0}^{m+n}i^{m+n-k}j^k[/tex]

    [tex]1=(i+j)^{mn}=\sum_{k=0}^{mn}{mn \choose k}i^{mn-k}j^k=\left(\sum_{k=0}^m {m \choose k}i^{m-k}j^k\right)^n[/tex]
    I can't see where that leads me. I understand that if I can write 1 as [itex]i^n x+j^m y[/itex] whatever x and y are, then I`m done.
     
  5. Sep 19, 2005 #4

    Galileo

    User Avatar
    Science Advisor
    Homework Helper

    I think I've got it.

    [tex]1=(i+j)^{m+n}=\sum_{k=0}^{m+n}{m+n \choose k}i^{m+n-k}j^k[/tex]
    [tex]\sum_{k=0}^{m}{m+n \choose k}i^{m+n-k}j^k +\sum_{k=m+1}^{m+n}{m+n \choose k}i^{m+n-k}j^k=[/tex]
    [tex]i^n\left(\sum_{k=0}^{m}{m+n \choose k}i^{m-k}j^k\right) +j^m\left(\sum_{k=0}^{n-1}{m+n \choose m+1+k}i^{n-1-k}j^{1+k}\right)[/tex]
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Algebra problem
  1. Algebra problem (Replies: 2)

  2. Lin algebra problem (Replies: 1)

Loading...