Left zero divisor, right inverse

[Grigorios Kostakos]

Examine if the following proposition holds:
There exists a non-zero element $r$ of a unitary ring $R$, such that $r$ is a left zero divisor and has left inverse, i.e. there exist $s,t\in R\setminus\{0_R\}$ such that $r\,s=0_R$ and $t\,r=1_R$.