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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).

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    What does "in this case" mean? What case? Did you mean to ask: "Suppose $A$ is a ring, and $a$ is a non-unit. Does there exist an extension $B$ of $A$ in which $a$ is a zero divisor?"2011-12-30
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    What precisely do you mean by "extension"? Is $A$ supposed to be a subring of $J$ with the same identity element? Otherwise you could take $J=A\times A$, and $A\to A\times 0$, in which case every element of $A$ becomes a zero divisor in $J$.2011-12-30

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