A commutativity study for periodic rings
Howard E. Bell · Pacific Journal of Mathematics · 1977
Putcha and Yaqub have proved that a ring R satisfying a polynomial identity of the form xy = ω (JC, y), where ω (X, Y) is a word different from XY, must have nil commutator ideal.Our first major theorem extends this result to the case where ω (X, Y) varies with x and y, with the restriction that all ω(X, Y) have length at least three and are not of the form X n Y or XY".Further restrictions on the ω(X, Y) are then shown to yield commutativity of JR among these is a semigroup condition of Tamura, Putcha, and Weissglass-sepecifically, that each ω (X, Y) begins with Y and has degree at least 2 in X.The final theorem establishes commutativity of rings R satisfying xy = yxs, where s = s(x, y) is an element in the center of the subring generated by x and y.All rings considered are either periodic by hypothesis or turn out to be periodic in the course of the investigation.