Ring of Integers Isomorphism Problem

In summary, to prove that ZN is isomorphic to ZA x ZB, we can use the map f(n) = (n mod A, n mod B) and the Chinese remainder theorem. This map is both homomorphic and injective, and since ZN and ZA x ZB have the same cardinality, we can conclude that f is surjective. However, if we are given an (a, b) in ZA x ZB, we may have difficulty finding an n such that f(n) = (a, b). In this case, we can use the Chinese remainder theorem to simplify the process.
  • #1
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Homework Statement
Let N = AB, where A and B are positive integers that are relatively prime. Prove that ZN is isomorphic to ZA x ZB.

The attempt at a solution
I'm considering the map f(n) = (n mod A, n mod B). I've been able to prove that it is homomorphic and injective. Is it safe to conclude, since ZN and ZA x ZB have the same cardinality and f is injective, that f is surjetive? In any case, given an (a, b) in ZA x ZB, I've been trying to find an n such that f(n) = (a, b) without success. Any tips?
 
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  • #2
Recall the Chinese remainder theorem.
 
  • #3
Good tip. Thanks.
 

Related to Ring of Integers Isomorphism Problem

1. What is the Ring of Integers Isomorphism Problem?

The Ring of Integers Isomorphism Problem is a mathematical problem that aims to determine whether two rings of integers are isomorphic, meaning that they have the same algebraic structure.

2. How is the Ring of Integers Isomorphism Problem solved?

The problem is typically solved using techniques from abstract algebra, such as group theory and ring theory. These techniques involve analyzing the properties and operations of the rings to determine if they are isomorphic.

3. Why is the Ring of Integers Isomorphism Problem important?

The problem has applications in various areas of mathematics, including number theory, algebraic geometry, and algebraic topology. It also has practical applications in cryptography and coding theory.

4. Are there any known solutions to the Ring of Integers Isomorphism Problem?

Yes, there are known solutions for specific cases of the problem, such as when the rings have special properties or when certain conditions are met. However, a general solution for all cases of the problem is still an open question.

5. What are some challenges in solving the Ring of Integers Isomorphism Problem?

One of the main challenges is that the problem is NP-hard, meaning that it is difficult to find an efficient algorithm to solve it for all cases. Another challenge is that the problem becomes more complex as the size of the rings increases, making it difficult to analyze and compare the structures of the rings.

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