MathNet 2010 — Part 05 Question 62

Let $x$ and $y$ be two elements of a ring $R$, such that $xyx \ne 0$ and $(xy)^n = 0$, where $n \ge 2$ is an integer. Show that there exists an element $z$ in $R$, such that $xz \ne 0$ and $xyxz = 0$.