Proving Subring: Odd Denominator Fractions in Q

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In summary, the rational numbers include all fractions with an odd denominator. The collection of all fractions is a subring of the rational numbers and can be proved to be a ring.
  • #1
silvermane
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Homework Statement


Inside the rational numbers Q, let's say we have a collection of all fractions where they all have an odd denominator. Prove that this collection is a subring of Q.

The Attempt at a Solution


I know that a ring is a set, R, with two operations, + and *, and two special elements, 0 and 1, that satisfy:

1.) R with the operation + is a group with identity 0, called the zero element of R. The element b so that a+b = b+a = 0 is the negative of a.
2.) R with the operation * satisfies the associative property, and the element 1 is the identity element under multiplication.
3.) R with + and * satisfies the distributive laws: for every a, b, c in R, a(b+c) = (ab)+(ac), and (a+b)c = (ac) + (bc).

I know these are some of the rules, but I don't know if I just have to show that these are true for our set {..., -1/5,-1/3, -1, 0 , 1, 1/3, 1/5, ...} or if there are more to show. Any helpful hints or tips would be well appreciated!

Thank you for your help in advance! :))
 
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  • #2
You don’t have to show all of those properties, for example you know * is associative because it’s elements come from a ring.

You just need to show it has the additive identity, additive inverses, it’s closed under addition, has multiplicative identity, and it’s closed under multiplication. Everything else comes from it already being part of a ring.
 
  • #3
JonF said:
You don’t have to show all of those properties, for example you know * is associative because it’s elements come from a ring.

You just need to show it has the additive identity, additive inverses, it’s closed under addition, has multiplicative identity, and it’s closed under multiplication. Everything else comes from it already being part of a ring.

Okay, that works. I just wanted to make sure I was on the right track. Thanks! :)
 

FAQ: Proving Subring: Odd Denominator Fractions in Q

1. What is a subring?

A subring is a subset of a ring that is itself a ring under the same operations as the original ring. In other words, a subring is a smaller ring that shares the same algebraic structure as the original ring.

2. How do you prove that a subset is a subring?

To prove that a subset is a subring, we need to show that it satisfies the three conditions for being a subring: closure under addition, closure under multiplication, and closure under additive inverse. This means that any two elements in the subset added together, multiplied together, or with their additive inverse, must also be in the subset. We also need to show that the subset contains the identity element of the original ring.

3. What are odd denominator fractions in Q?

Odd denominator fractions in Q are fractions where the denominator is an odd number. In other words, the denominator cannot be divided evenly by 2.

4. Why is it important to prove that odd denominator fractions in Q form a subring?

Proving that odd denominator fractions in Q form a subring is important because it helps us understand the algebraic structure of this subset of rational numbers. It also allows us to make certain mathematical statements and proofs more concise and efficient by using the properties of a subring.

5. Can you give an example of a proof for proving odd denominator fractions in Q as a subring?

Yes, for example, we can prove that the set of odd denominator fractions in Q is closed under addition by showing that if we take any two odd denominator fractions, their sum will also have an odd denominator. We can do this by expressing the fractions in terms of their prime factorization and showing that the denominator of the sum has all the same prime factors as the individual denominators. Similarly, we can prove closure under multiplication and additive inverse using similar methods.

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