4. In constructing the field of fractions of an integral domain R, prove that multiplication of equivalence classes is well defined: that is if [(a, b)] = [(c,d)] and [(a', b')] = [(c', d')], then [(aa', bb')] = [(cc',dd')] =
4. In constructing the field of fractions of an integral domain R, prove that multiplication of equivalence classes is well defined: that is if [(a, b)] = [(c,d)] and [(a', b')] = [(c', d')], then [(aa', bb')] = [(cc',dd')] =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem 4: Field of Fractions of an Integral Domain**
In constructing the field of fractions of an integral domain \( R \), prove that multiplication of equivalence classes is well defined. Specifically, demonstrate that if \([(a, b)] = [(c, d)]\) and \([(a', b')] = [(c', d')]\), then \([(aa', bb')]\) is equal to \([(cc', dd')]\).
**Explanation:**
This exercise involves proving the consistency of defining multiplication in the field of fractions. The field of fractions of an integral domain allows us to create a field where arithmetic operations are performed on equivalence classes of ordered pairs. Here, multiplication must respect the equivalence relation, ensuring that the operation is well-defined.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c55fd55-ae67-4b97-a36c-91359ff73a6f%2F1fd4c1f0-c32d-400d-8c87-4def71a33f76%2Fodvhfi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 4: Field of Fractions of an Integral Domain**
In constructing the field of fractions of an integral domain \( R \), prove that multiplication of equivalence classes is well defined. Specifically, demonstrate that if \([(a, b)] = [(c, d)]\) and \([(a', b')] = [(c', d')]\), then \([(aa', bb')]\) is equal to \([(cc', dd')]\).
**Explanation:**
This exercise involves proving the consistency of defining multiplication in the field of fractions. The field of fractions of an integral domain allows us to create a field where arithmetic operations are performed on equivalence classes of ordered pairs. Here, multiplication must respect the equivalence relation, ensuring that the operation is well-defined.
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