Everyone knows that if in a ring A a unit a $\in$ A can´t be a zerodivisor. But could also be possible that "a" not be a zero divisor ( i.e does not exist a nonzero x $\in$ A , such that $ax=0$) but neither a unit ( in A ) in this case
My question is so simple in this case, considering the ring of fractions, we know that there exist an extension such that "a" is a unit . My question is, in this case, there exist other extension $J$ of $A$ , such that $a$ is a zero divisor in J ( i.e there exist a nonzero x $\in$ $J$ , such that $ax=0$) . So the question is obviously false if we consider "a" as a unit, but I think that here could be true.
Remark : I think that we may assume that A is commutative, but I´m not completely sure ( because we use the ring of fractions the commutative property is needed otherwise the sum in the fraction won´t be commutative). If you note that other properties are also needed please let me know ( like identity , etc).