The common methods for proving $A = B$ and $A \approx B$ are very different. To prove a set equality $A = B$, the common method is to show that $A \subseteq B$ and $B \subseteq A$. For the former, we show that for any $a \in A$, $a$ also has properties which guarantee that $a \in B$, and likewise for the latter. Showing that two arbitrary rings $A \approx B$ is an entirely different process; generally a one-to-one, onto, and operation preserving isomorphism $\phi: A \to B$ must be found.
Importantly, it seems as though it is impossible to 'swap' these proof techniques. It doesn't make sense, to prove, say,
$$Z_{10} \approx Z_5 \otimes Z_2$$
By trying to show $Z_{10} \subset Z_5 \otimes Z_2$ and $Z_5 \otimes Z_2 \subset Z_{10}$, because, if interpreting these symbols literally, this is nonsensical. Showing that some $(a, b) \in Z_5 \otimes Z_2$ exists also in $Z_{10}$ makes no sense because the elements and operations don't match.
Which is why I'm wondering that something like this seems to happen often in proofs relating to fields of quotients.
For example, the question which I encountered which started this whole confusion.
Let $Z[i]$ = $\{a + bi | a, b \in \mathbb{Z}\}$. Show that the field of quotients of $Z[i]$, $F$, is isomorphic to $Q[i] = \{a + bi | a, b \in \mathbb{Q}, b \ne 0\}$
This question has been asked on the Maths Stack Exchange before,and has been answered here.
The rough way that the proof goes, according to this answer (as well as the slightly different answer in the textbook itself) is as follows:
First, show that $Q[i] \subseteq F$. This is done by showing some $\frac{a}{b} + \frac{c}{d}i \in Q[i]$ can also be represented as a single fraction, $\frac{ad + cbi}{bd}$. It is now in the form of an element of $F$.
Second, show that $F \subseteq Q[i]$. This is done by showing some $\frac{a + bi}{c + di} \in F$ can, by multiplying by the conjugate, be represented like so:
$$\frac{ac + bd + (bc - ad)i}{c^2 + d^2} = \frac{ac + bd}{c^2 + d^2} + \frac{bc-ad}{c^2 + d^2}i \in Q[i]$$
At first glance this seems convincing, and I am sure that it is correct--but the more I seem to think about it, the less it makes sense. $F$, the field of quotients, is a set of equivalency classes, whilst $Q[i]$ is simply a subset of $\mathbb{C}$. Isn't proceeding by the proof above, showing that $F \approx Q[i]$ by showing $Q[i] \subseteq F$ and $F \subseteq Q[i]$ just as absurd as the earlier example of showing $Z_{10} \approx Z_5 \otimes Z_2$ by showing $Z_{10} \subset Z_5 \otimes Z_2$ and $Z_5 \otimes Z_2 \subset Z_{10}$?
Phrased another way, how can we rigorously say that (in the first part of the proof)
$$\frac{ad + cbi}{bd} \in F$$
When the element on the left-hand side is some "number" from $Q[i] \subset \mathbb{C}$ and the right hand side is a set of equivalency classes? Isn't this as absurd as the statement "$(a, b) \in Z_{10}$"?
(A similar, but opposite, argument holds for the second part of the proof).
