I am reading Dummit and Foote Section 7.5 Rings of Fractions - and in particular Theorem 15 - see attached.

I am at ease with the proof of Theorem 15 (see attachment) down to where D&F state near the bottom of page 262:

"Next we embed R into Q by defining

by "

D&F then state that i is a ring homomorphism

I am OK with i(a + b) = i(a) + i(b) for a, b Q since

i(a+b) =

However, my problem is that I am slightly uncomfortable with demostrating that i(ab) = i(a)i(b) since we have

{is this actually correct?}

and

So it looks as if we need to demostrate that

but why (exactly!!) is this the case

Now we could write

It looks as if ... but why exactly??

Can anyone clarify this for me?

Peter