Abstract Algebra: Ring Isomorphism Construction

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Homework Help Overview

The discussion revolves around the construction of ring isomorphisms in the context of abstract algebra, specifically relating to the Chinese remainder theorem and properties of ideals in rings.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of the Chinese remainder theorem and its application to ring isomorphisms. There are attempts to prove certain properties of ideals, such as the sum of ideals equating to the ring itself. Questions arise regarding the correctness of notation and the validity of specific steps in the proof.

Discussion Status

Some participants have provided insights into the proof structure and notation, while others are questioning specific aspects of the reasoning. There is an ongoing exploration of the relationships between elements in the ideals and their images under the proposed function.

Contextual Notes

There are indications of potential notation errors that participants are addressing, which may affect the clarity of the arguments presented. The discussion is framed within the constraints of homework guidelines, focusing on understanding rather than providing direct solutions.

lola1990
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Homework Statement


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The Attempt at a Solution

 
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Do you know the chinese remainder theorem??
 
What bothers you about it??
 
lola1990 said:
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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