Functional Identities in (Semi)prime Rings
Up until now we have seen how to construct new d-free sets from given ones (Chapter 3) and have analyzed certain functional identities acting on d-free sets (Chapter 4). But, with the exception of the results from Chapter 2, we have yet to show the existence of important classes of d-free sets. Our main purpose in this chapter is to remedy this situation. Our success in this endeavor has chiefly been in the case of various subsets of a prime ring A (considered as a subring of its maximal left quotient ring Q). These results will be presented in Section 5.2. They will be obtained as corollaries to the results from Section 5.1 which establish the d-freeness of rings that contain elements satisfying certain technical conditions — specifically, the so-called fractional degree of such elements must be ≥ d. In Section 5.3 we shall see that the basic result on d-freeness of prime rings can be extended to a more general (and truly more entangled) semiprime setting. Section 5.4 is devoted to commuting traces of multiadditive maps on prime rings; the definitive result is established in the case of biadditive maps. The chapter ends with Section 5.5 which studies generalized functional identities in prime rings.
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