Commutativity of rings with constraints involving a subset

Khan, Moharram A.

Czechoslovak Mathematical Journal, Tome 53 (2003) no. 3, pp. 545-559.

Résumé

Suppose that $R$ is an associative ring with identity $1$, $J(R)$ the Jacobson radical of $R$, and $N(R)$ the set of nilpotent elements of $R$. Let $m \ge 1$ be a fixed positive integer and $R$ an $m$-torsion-free ring with identity $1$. The main result of the present paper asserts that $R$ is commutative if $R$ satisfies both the conditions (i) $[x^m,y^m] = 0$ for all $x,y \in R \setminus J(R)$ and (ii) $[(xy)^m + y^mx^m, x] = 0 = [(yx)^m + x^my^m, x]$, for all $x,y \in R \setminus J(R)$. This result is also valid if (i) and (ii) are replaced by (i)$^{\prime }$ $[x^m,y^m] = 0$ for all $x,y \in R \setminus N(R)$ and (ii)$^{\prime }$ $[(xy)^m + y^m x^m, x] = 0 = [(yx)^m + x^m y^m, x]$ for all $x,y \in R\backslash N(R) $. Other similar commutativity theorems are also discussed.

Détail

Classification : 16R50, 16U70, 16U80, 16U99
Keywords: commutativity theorems; Jacobson radicals; nilpotent elements; periodic rings; torsion-free rings