Proving Subring: Odd Denominator Fractions in Q

[silvermane]

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!

 

[JonF]

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.

 

[silvermane]

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! :)