suppose rd^{-1}is in ker(Φ).

this means Φ(rd^{-1}) = φ(r)φ(d)^{-1}= 0_{S}.

thus φ(r) = φ(r)(1_{S}) = φ(r)(φ(d)^{-1}φ(d)) = (φ(r)φ(d)^{-1})φ(d) = (0_{S})φ(d) = 0_{S}.

hence r is in ker(φ). thus (since φ is injective, r = 0_{R}) and thus rd^{-1}= 0_{Q}

(the equivalence class of (0,d) is indeed the additive identity of Q = D^{-1}R).

**********

now, suppose T is a ring such that: T contains R as a subring, and every d in D (which is a subset of R) is a unit of T.

since T contains R as a subring, the inclusion mapping i:R→T given by i(r) = r, is clearly an injective ring homomorphism.

moreover, i(d) = d is a unit in T.

we thus get an injective ring homomorphism I:Q→T. thus I(Q) is a subring of T (ring)-isomorphic to Q.

it is common practice to "identify" isomorphic structures (just as we regard the rational number n/1 as "the integer" n).

Q is only unique, UP TO ISOMORPHISM. that is what the proof shows: if we have a ring S that embeds R and in which all elements of D are units, it has a subring isomorphic to Q.

that is, if the mapping Φ in D&F's proof is surjective, it is an isomorphism.

this is "common" in abstract algebra: since one is only concerned with classifying algebraic objects up to isomorphism type (which, being an equivalence, is "almost as good as equality") one often sees the following definitions:

a subobject B (up to isomorphism) of an object A, is an injective homomorphism from B to A.

a quotient object C (up to isomorphism) of an object A is a surjective homomorphism from A to C.

(note that these definitions only hold for "certain kinds" of objects: most notably- sets, groups, rings, modules, and vector spaces).

the basic idea is: it turns out that all the "interesting stuff" in algebra isn't really in the objects themselves, it's happening in the "structure-preserving maps" BETWEEN objects.

if you want to understand groups, you study homomorphisms. if you want to understand vector spaces, you study linear transformations. if you want to understand sets, you look at functions.

it is a bit unsettling to think of a sub-something as just "an injective homomorphism". as humans, we're more "object-oriented", and not "process-oriented". but that is where the abstraction leads, it turns out that the properties of "object A" are better described by "isomorphically preserved information" than explicit, intrinsic description.

for example, what is relevant about Z_{4}is that it is a cyclic group of order 4, it really "doesn't matter" (as far as GROUP theory is concerned) if we are talking about:

{0,1,2,3} under addition mod 4

{1,i,-1,-i} under complex multiplication <---aka the roots of x_{4}- 1 (see last example).

{e, (1 2 3 4), (1 3)(2 4), (1 4 3 2)} under functional composition

, under matrix multiplication

{e,x,x^{2},x^{3}; x^{4}= e}

the group information that is pertinent is the same for all of these.