Let $I\not = (0)$ be an ideal without non-zero divisors in a ring $R.$ If there exists non-zero $a\in I$ and $r\in R$ such that $ra=0$ then $rI= Ir=(0).$
My Attempt: If $r=0$ then $rI=Ir=(0)$ and we have nothing to prove. If $r\not =0$ then $ra=0$ means that $a$ and $r$ are zero-divisors. So now we want to show that $$rx=0$$ for all $x\in I.$ After this I am not sure what to do. Any hints will be much appreciated.
