Confusion about the definition of localization

Question

For example, let $\mathbb Z$ be the ring and $S = \mathbb Z - 2\mathbb Z$. Then the quotient ring should be:
$S^{-1}A = \{a/s: a\in \mathbb Z \text{ and }s \in S\}$, which is formed by the equivalence class $(a,s) \sim (a',s')$ if $\exists t\in S$ s.t. $(as'-a's)t = 0$.
My question:

Can we say $5/3 = 10/6$?

I don't think we can since $6\notin S$, but I may see things like appeared before.

Answer 1

As you say: $10/6$ is not in your ring.