Derivations cocentralizing multilinear polynomials (Q1365328)

The main result of this paper shows that if \(R\) is a prime \(K\) algebra, \(f(X_1,\dots,X_n)\) is a multilinear and homogeneous polynomial over \(K\) in noncommuting variables, \(D\) and \(E\) are derivations of \(R\), and \(D(f(y_1,\dots,y_n))f(y_1,\dots,y_n)-f(y_1,\dots,y_n)E(f(y_1,\dots,y_n))\) is central for all \(y_i\in I\), a nonzero ideal of \(R\), then either \(D=E=0\), or \(E=-D\) and \(f(X_1,\dots,X_n)^2\) is central valued on \(R\), except when \(\text{char} R=2\) and \(R\) satisfies the standard identity \(S_4\).