Null element in a ring

Question

In an arbitrary ring $A$ (not necessarily unitary) , I have an $a\in A$ such that $ab=0 $ $\forall b\in A$. How can I prove $a=0$?

EDIT: If the result is false for non-unitary rings, please give a counterexample

Answer 1

Take any abelian group $A$ with more than one element and define $a\cdot b = 0$ for all $a, b \in A$. This yields a non-unitary ring with trivial multiplication. It is a counterexample as desired.

Answer 2

It's not necessarily true for a non-unitary ring.

Consider the ring consisting of the two integers $\{0, 6\}$ with addition and multiplication both modulo 12 as a counterexample.