FUNCTIONAL IDENTITIES REVISED: THE FRACTIONAL AND THE STRONG DEGREE

ABSTRACT

The concepts of the fractional and the strong degree of an element in a ring are introduced. It is shown that definitive results on functional identities can be obtained in rings which contain elements of appropriate fractional (or strong) degree. This enables us to extend the results on functional identities from prime to semiprime rings, as well as to some rather different classes of rings, such as matrix rings over any unital ring. As an application, commuting maps, Lie derivations and commutativity preserving maps in such rings are discussed.