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Math Help - relatively prime ideals

  1. #1
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    relatively prime ideals

    Suppose R is a ring and I, J are relatively prime ideals of R; that is, I + J = R. Prove that for any integers n, m \leq 1, I^n+J^m=R.
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  2. #2
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    Quote Originally Posted by Erdos32212 View Post
    Suppose R is a ring and I, J are relatively prime ideals of R; that is, I + J = R. Prove that for any integers n, m \leq 1, I^n+J^m=R.
    first of all, that should be m,n \geq 1, and also i assume that your ring is commutative with unity. so you only need to show that 1 \in I^n +J^m. this is very easy:

    since I+J=R, there exist r \in I, \ s \in J such that r+s=1. thus: 1=(r+s)^{m+n}=r^n \sum_{i=0}^m \binom{n+m}{i}r^{m-i}s^i \ + \ s^m \sum_{i=m+1}^{n+m} \binom{n+m}{i}r^{n+m-i}s^{i-m} \in I^n + J^m. \ \Box
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