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ring arithmetic, x^n=x for all x implies x,y^(n-1) commute

Source: VJIMC 2001 2.4

July 21, 2021
Ring Theoryabstract algebra

Problem Statement

Let RR be an associative non-commutative ring and let n>2n>2 be a fixed natural number. Assume that xn=xx^n=x for all x∈Rx\in R. Prove that xyn−1=yn−1xxy^{n-1}=y^{n-1}x holds for all x,y∈Rx,y\in R.
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