Vol. 70, No. 1, 1977

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A commutativity study for periodic rings

Howard Edwin Bell

Vol. 70 (1977), No. 1, 29–36

Putcha and Yaqub have proved that a ring R satisfying a polynomial identity of the form xy = ω(x,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 XnY or XY n. Further restrictions on the ω(X,Y ) are then shown to yield commutativity of R; 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.

Mathematical Subject Classification
Primary: 16A70, 16A70
Received: 9 December 1976
Revised: 3 March 1977
Published: 1 May 1977
Howard Edwin Bell