# 2009-16 Commutative ring

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.

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