= Total ring of fractions
{title2=$Q(A)=S^{-1}A$}
{wiki}
= Total quotient ring
{synonym}
Here $S$ consists of the <non-zero-divisors> of a <commutative ring> $A$. The <localization> map $A\to Q(A)$ is <injective>: a fraction $a/1$ is zero only if some $s\in S$ annihilates $a$. For a <reduced ring> with finitely many <minimal prime ideals>, its <zero divisors> form the union of these primes. Indeed, if $ab=0$ and $a$ avoids every minimal prime, reduction to each domain quotient forces $b$ into their intersection, which is zero. Conversely, for $a$ in a minimal prime $\mathfrak p_i$, choose $b$ in every other minimal prime but outside $\mathfrak p_i$, using their incomparability and multiplying suitable elements. Then $b\ne0$ and $ab=0$.
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