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 XYn. 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.
