# 2012-16 A finite ring

Prove that if a finite ring has two elements $$x$$ and $$y$$ such that $$xy^2=y$$, then $$yxy=y$$.

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Prove that if $$a_1 a_2\ldots a_k=a_{\pi(1)} a_{\pi(2)} \ldots a_{\pi(k)}$$  for any elements $$a_1, a_2,\ldots,a_k \in I$$, then R is commutative.