Abstract Algebra: Ring Isomorphism Construction

  • Thread starter lola1990
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  • #1
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Homework Statement





Homework Equations





The Attempt at a Solution

 
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Answers and Replies

  • #2
22,129
3,297
Do you know the chinese remainder theorem??
 
  • #3
22,129
3,297
What bothers you about it??
 
  • #4
22,129
3,297
So I already proved that I+J=R, so that there are x in I and y in J such that x+y=1. Then, f(x)=(I, 1+J) because x is in I and x=1-y which is in 1+B. Similarly, f(y)=(1+I, J). Now, consider kx+ry. f(kx+ry)=f(k)f(x)+f(r)f(y)=(k+I, k+J)(I, 1+J)+(r+I, r+J)(1+I, J)=(I, k+J)+(r+I)(J)=(r+I)(k+J). Is that right?

The bolded part have a wrong notation. It must be (r+I,J) and (r+I,k+J). But it is correct.
 

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