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?