Are f.g. projective modules (of constant rank) free over the ring $A$ which is the total quotient ring of a reduced nonNoetherian commutative ring. Note that dimension of $A$ need not be $0$.
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I've posted an answer here. 


I can give an example of ring $A$ which is reduced and every nonunit is a zerodivisor. It may be helpful to find the answer over $A$. Let $R=k[x_1,x_2,...]$ be polynomial ring (in infinitely many variables) over a field; let $m =(x_1,x_2,...)$ be a maximal ideal. Fix an integer $n>0$ and define $I$ to be the ideal generated by products $x_l x_j$ with $l$, $j$ distinct and $l> n$. Define $A= R_m /I$. Then $ A$ is reduced, total quotient field of $A$ is $A$, and dimension of $A$ is $n$. The dimension of a reduced commutative Noetherian ring having no nonzerodivisor is $0$. Hence Tom's comment will not work in general, as commented by others. 

