Why does one want to have the standard definition of localization? The standard way of defining the localization of a commutative ring is as follows: given a multiplicatively closed subset $S\subset R$ the localization is defined first by considering the set $R\times S/\sim$ where
$$
(r,s) \sim (r',s') \text{ if there exists a } u \in S \text{ such that } u(rs' - r's) =0
$$
the rest is just equipping this set with a ring structure, but my question lies here: why do we want the $u$? Don't we not want nilpotents in our denominator? 

If you don't require the existence of such an $u$, $\sim$ won't be transitive in general (and hence not an equivalence relation). This becomes quite apparent when you attempt to prove that transitivity holds.
