Abstract Algebra: Ring Isomorphism Construction

[lola1990]

Homework Statement

Homework Equations

The Attempt at a Solution

 

[micromass]

Do you know the chinese remainder theorem??

 

[micromass]

What bothers you about it??

 

[micromass]

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.