On commutativity of P.I. rings (Q792433)

The author proves the commutativity of some classes of rings satisfying a polynomial identity \([X,Y]=f(X,Y)\), which does not hold for total matrix algebras over a field. Here f(X,Y) belongs to the commutator ideal of the free associative algebra over a commutative ring F. In particular, if \(\phi(X),\psi(X)\in X^ 2F[X],\) then the identity \([X,Y]=[\phi(X),Y]+[\psi(Y),X]\) yields commutativity.