A quesion about Field of quotients

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In summary, the question is asking for an explanation of why a commutative ring with unity that is not an integral domain cannot be contained in a field. The answer involves comparing the ring to the field of quotients and showing that such a situation would lead to a contradiction.
  • #1
wowolala
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The question is to explain why a commutative ring with unity that is not an integral domain cannot be contained in a field.

actually, every field is an integral domain. but above question wants us to compare with thm of Field of quotients..


first , i suppose the R is contained in a field, therefore by thm, can i conlude that R is isomorphic to an integral domain D, moreover, R is an integral domain, so we have contradicton with assumption..


am i right?


can anyone help me ?

thx so much
 
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  • #2
If you could embed R into a field F, then take a, b in R such that ab = 0 but a and b are nonzero. Then they have the same property in F, which is bad. So your explanation is fine.
 

Related to A quesion about Field of quotients

1. What is the Field of Quotients?

The Field of Quotients is a mathematical concept that is used to extend the notion of fractions to other mathematical structures, such as rings and integral domains. It is essentially a way to create a field from a given integral domain.

2. What is the purpose of the Field of Quotients?

The Field of Quotients is useful for several reasons. It allows us to perform division on elements in an integral domain, which is not always possible. It also allows us to generalize the concept of fractions to other mathematical structures, providing a deeper understanding of these structures.

3. How is the Field of Quotients constructed?

The Field of Quotients is constructed by taking the elements of an integral domain and creating equivalence classes of fractions, where the numerator and denominator are elements of the domain. These fractions are then defined by a set of equivalence rules, resulting in a new mathematical structure that is a field.

4. What are some examples of the Field of Quotients?

One example of the Field of Quotients is the field of rational numbers, which is constructed from the integral domain of integers. Another example is the field of real numbers, which is constructed from the integral domain of polynomials with real coefficients.

5. How is the Field of Quotients used in other areas of mathematics?

The Field of Quotients has various applications in mathematics, including algebraic number theory, algebraic geometry, and commutative algebra. It is also used in the construction of field extensions and in solving equations with rational coefficients.

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