In a ring R, if there exists a in R s.t. ab=0 for any b in R, then a must be 0. Is this statement true?

Question

In a general ring $R$, if there exists $a$ in $R$ s.t. $ab=0$ for any $b$ in $R$, then $a$ must be $0$. Is this statement true?

Answer 1

That depends on your definition of a ring. If $R$ is a ring with unity, then yes: we may simply select $b = 1_R$, and the result follows. If not, we can take the subring $\{0,2\}$ of $\mathbb{Z}_4$, and note that $2$ is such an element.