SCHWEITZER 1983 Problem 2

Let $I$ be an ideal of the ring $R$ and $f$ a nonidentity permutation of the set $\{ 1,2,\ldots, k \}$ for some $k$ . Suppose that for every $0 \not= a \in R, aI \not= 0$ and $Ia \not=0$ hold; furthermore, for any elements $x_1,x_2,\ldots ,x_k \in I$ , $$x_1x_2\ldots x_k=x_{1f}x_{2f}\ldots x_{kf}$$ holds. Prove that $R$ is commutative.

R. Wiegandt