I am reading Dummit and Foote, section 7.5 Rings of Fractions. I am working through Theorem 15 on page 261 (see attachment)

I am happy with D&F's prrof of Th 15 (thanks in part to a post by Deveno) down to the following paragraph: (see attachment)

"Next note that each has a multiplicative inverse in Q: namely if d is represented by the fraction then its multiplicative inverse is . One sees that every element of Q can be written as for some and some ."

Presumably in this paragraph so that we are sure that . We can also be sure that the inverse is unique because if d is represented by with inverse where then ("cross multiply")

But when D&F write:

"One sees that every element of Q can be written as for some and some ."- what exactly do they mean by - is this just shorthand for ?

Can someone please clarify this for me?

Peter