Effect of Polynomial Identity x[x, y] = (x[x, y])n in the Commutativity of Rings

Zohre Tabatabaei

Abstract


In this paper we study some sufficient conditions for commutativity of a ring according to Jacobsons'idea. Jacobson proved that if R isaringsatisfying xn =x (n>1) foreach x∈R, then R is commutative. In this paper, we show that R is commutative if for every x, y ∈ R there exists a positive integer n = n(x, y) such that (x[x,y])n =x[x,y].

Keywords


Commutator, left(right) s-unital, left semisim- ple ring, Jacobson Radical, left Primitive ring, division ring, faithful simple left R-module.

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