On almost-quasi-commutative rings.
Let \(R\) be an associative ring and for any elements \(x,y\in R\) there exist a nonnegative integer \(n=n(x,y)\) and nonzero relatively prime integers \(j(x,y)\) and \(k(x,y)\) such that \(jxy=k(yx)^n\). The authors prove the following main results: 1) the commutator ideal of \(R\) is nil, any idempotent of \(R\) is central; 2) if \(R\) is torsion-free and for any \(x,y\in R\) \(n(x,y)\geq 2\) then \(R\) is commutative; 3) if \(R\) is torsion-free and for any \(x,y\in R\) \(n(x,y)=1\) then \(R\) is either commutative or nil; 4) if the set of all nilpotent elements of a periodic ring \(R\) is commutative and \(n=n(x,y)\geq 1\) is a fixed number for any \(x,y\in R\) then \(R\) is commutative; 5) if \(R\) is a ring with \(1\) and for any \(x,y\in R\) \(n=n(x,y)\) is a fixed integer and either \(n\leq 2\) or \(n\geq 3\) and \(R\) is \((n-1)\)-torsion-free then \(R\) is commutative.
scientific article; zbMATH DE number 5651861