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 $\displaystyle d \in D$ has a multiplicative inverse in Q: namely if d is represented by the fraction $\displaystyle \frac{de}{e} $ then its multiplicative inverse is $\displaystyle \frac{e}{de}$. One sees that every element of Q can be written as $\displaystyle r \cdot d^-1 $ for some $\displaystyle r \in R $ and some $\displaystyle d \in D $."

Presumably in this paragraph $\displaystyle e \in D $ so that we are sure that $\displaystyle de \in D $ . We can also be sure that the inverse is unique because if d is represented by $\displaystyle \frac{df}{f}$ with inverse $\displaystyle \frac{f}{df}$ where $\displaystyle f \in D $ then $\displaystyle \frac{f}{df} \sim \frac{e}{de} $ ("cross multiply")

But when D&F write:

"One sees that every element of Q can be written as $\displaystyle r \cdot d^{-1} $ for some $\displaystyle r \in R $ and some $\displaystyle d \in D $."- what exactly do they mean by $\displaystyle r \cdot d^{-1}$ - is this just shorthand for $\displaystyle \frac{re}{de} $?

Can someone please clarify this for me?

Peter