Rings of fractions isomorphic to rationals in Lovett 6.2.4

[Math Amateur]

I am reading Stephen Lovett's book, "Abstract Algebra: Structures and Applications" and am currently focused on Section 6.2: Rings of Fractions ...

I need some help with some remarks following Definition 6.2.4 ... ... ...

The remarks following Definition 6.2.4 reads as follows:

In the above text from Lovett we read the following:

" ... ... it is not hard to show that if we had taken $D = { \mathbb{Z} }^{ \gt 0 }$ we would get a ring of fractions that is that is isomorphic to $\mathbb{Q}$. ... ... "Can someone please help me to understand this statement ... how is such an isomorphism possible ... in particular, how does one achieve a one-to-one and onto homomorphism from the positive integers to the negative elements of $\mathbb{Q}$ as well as the positive elements ...

Hope someone can help ... ...

Peter==============================================================================

To enable readers to understand Lovett's approach to the rings of fraction construction, I am providing Lovett Section 6.2 up to an including the remarks following Definition 6.2.4 ... as follows:

 

[willem2]

There is no isomorphism between D and Q, but an isomorphism between a set of equivalence classes of pairs (r,d) (where r is in R and d is in D ) and Q.
The equivalence class containing all pairs (-n,2n) will map to -1/2, for example.

 

[Math Amateur]

Hi willem2

Thanks for the help ...

Obviously I should have read the text more carefully ...

Thanks again ...

Peter

 

[WWGD]

Should be Lovett and Leavitt or Lovett and Leavitt ;).