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 finite ring has two elements \(x\) and \(y\) such that \(xy^2=y\), then \( yxy=y\).

Let k>1 be a fixed integer. Let π be a fixed nonidentity permutation of {1,2,…,k}. Let *I* be an ideal of a ring R such that for any nonzero element a of R, a*I*≠0 and *I*a≠0 hold.

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.