I am working through Theorem 15 (Rings of Fractions) of Dummit and Foote Chapter 7 (see attachment)

I am unable to completely follow the details of the argument for the uniqueness property of Q - see page 263 of D&F (see attachment) where D&F's argument is as follows:

"It remains to establish the uniqueness propoerty of Q.

Assume is an injective ring homomorphism such that is a unit in S for all .

Extend to a map by defining for all .

The map is well defined since implies , so and then

.

It is straightforward to check that is a ring homomorphism - details left as an exercise.

Finally, is injective because implies

Since is injective ths forces r and hence to be zero.

This completes the proof"

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My problems are as follows:

1. I cannot follow the argument that implies

Can someone please give an explicit and formal set of steps that establishes this.

2. in the statement of Theorem 15 on page 261 (see attachment) D&F state that Q is unique and that the ring Q is the "smallest" ring containing R in which all elements of D become units

Can someone please explain how the above proof actually establishes these facts.

Peter