Localization at finitely many minimal prime ideals

Solution 1:

1) Let $\phi : A \to \prod_{i=1}^n A_{p_i}$ be the natural map (i.e. product of localization maps). Every element of $S$ is sent to a unit under $\phi$, so there is an induced map $\varphi : S^{-1}A \to \prod_{i=1}^n A_{p_i}$. But $\varphi$ is locally an isomorphism (at every maximal ideal $p_i$ of $S^{-1}A$, $\varphi_{p_i} : (S^{-1}A)_{p_i} \cong A_{p_i} \to (\prod_{i=1}^n A_{p_i})_{p_i} \cong A_{p_i}$), so $\varphi$ is globally an isomorphism.

2) For a reduced ring, the set of nonzerodivisors is precisely the complement of the union of the minimal primes (this is easy to see in the Noetherian case, but also holds in general). Thus in this case the two notions of total ring of fractions coincide.