Rings with permutable products
Let R be an associative ring and p(R) is the smallest \(n>1\) such that for any \(x_ 1,...,x_ n\in R\) there exists a non-identity permutation \(\sigma\) for which \[ x_ 1x_ 2...x_ n=x_{\sigma (1)}x_{\sigma (2)}...x_{\sigma (n)} \] (otherwise, p(R) is \(\infty)\). The authors prove the following results: i) if p(R)\(\leq 5\) then the commutator ideal C(R) is nil; (ii) if R is a 2-torsion-free ring with 1 and \(p(R)=3\) then R is commutative; iii) if R is an algebra over a field and \(p(R)<\infty\) then C(R) is nil; iv) \(p(M_ 2(GF(q)))\leq q\) \(3+q+2\); v) if \(n\geq 3\) and for any \(x_ 1,...,x_ n\in R\) there exists a fixed-point-free permutation \(\sigma\) such that \[ x_ 1x_ 2...x_ n=x_{\sigma (1)}x_{\sigma (2)}...x_{\sigma (n)}, \] then C(R) is nil.
scientific article; zbMATH DE number 4053851