Can zero divisors be in the denominator when we localize rings? Can we localize rings with zero divisors? Can those zero divisors be in the denominator?
I thought defining $$\frac{a}{b}=\frac{c}{d} \text{ iff }t(ad-bc)=0 \text{ where $b,d,t$ belong to the same multiplicative system}$$
accommodated for that little detail. But my professor thinks not. I am confused. 
Any help would be greatly appreciated.

Your definition is correct and even necessary: if you don't include the factor $t$ into the definition, then you will in general not get an equivalence relation between pairs (a,b) of ring elements. This however is necessary to define the notion of a fraction a/b.
