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, aI≠0 and Ia≠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.