Changing the zero product property and defining division by zero

Question

I know that defining division by zero is not possible because it violates the zero product property we define, that is, $0\times a=0$ for every $a$. I wonder whether we can somewhat circumvent and change the definition of the zero product property (or make a new definition about multiplying by zero), then safely deducing a definition for dividing by zero and then possibly building a system when this definition exists and proving useful theorems?