KAIST POW 2012-16 : A finite ring

A finite ring

 Prove that if a finite ring has two elements $x$ and $y$ such that $xyy=y$, then $yxy=y$.

 
 Proof. Consider a subset $\{y^k | k\in \mathbb{N} \}$. Since the ring is finite, the subset is also finite, so there must exist $n$,$m\in \mathbb{N}$ such that $y^n=y^{n+m}$. If $n>1$, $$y^{n-1}=xy^{n}=xy^{n+m}=y^{n-1+m}$$ By repeating this, we get $y=y^{m+1}$. Therefore, $$yxy=yxy^{m+1}=yy^m=y^{m+1}=y$$