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An exposition on a mathematical subject usually starts with basic definitions. We feel, however, that it is more appropriate to introduce functional identities through examples. We shall therefore present various simple examples of functional identities, so that the reader may guess which conclusions could be derived when facing these identities. These examples have been selected in order to illustrate the general theory, and not all of them are of great importance in their own right. Their consideration will be rather elementary; anyhow, many of the arguments that we shall present here will be used, sometimes in a hidden way, in much more general situations considered in further chapters. Examples will be followed by some basic definitions and notation, but even these will be given in a somewhat informal fashion. The last objective of this preliminary chapter is to point out a few instances where functional identities appear naturally. That is, we wish to indicate, without many details, why and where the theory of functional identities is applicable. So, in summary, the goal of this chapter is to give an informal introduction to functional identities which should be of help to a newcomer to the subject.

In Section 2.1 we will introduce the concept of the strong degree of a unital ring. The definition involves a condition which is rather technical, but we shall see that the strong degree can be rather easily computed for certain classes of rings. The main reason for dealing with this concept is its connection with functional identities - this will be the topic of Section 2.4. Before that, in Sections 2.2 and 2.3, we will consider certain versions of the concept of -freeness (called strong -freeness and strong (; )-freeness). Unlike in Chapter 1, we shall now consider these notions in a rigorous manner.

In this chapter we will study -free sets and some related (but more complicated) concepts such as (; )-freeness and (*; ; )-freeness. Besides introducing these concepts and considering their formal properties, our main objective will be to present various constructions that yield new -free (resp. (; )-free) sets from given ones. In Chapter 5 we shall actually establish -freeness of certain particular classes of sets and then use the results of the present chapter to show that the list of -free sets is really quite extensive.

Chapter 3 was primarily devoted to constructing new -free sets from given -free or (; )-free sets. Now we turn our attention to the study of FI’s on -free sets. Of course, by the very definition one can say everything that is possible about the basic FI’s through which -free sets were introduced. But what we intend to show is that one can analyze also some other FI’s on -free sets, some of them considerably more complicated than the basic ones.

Up until now we have seen how to construct new -free sets from given ones (Chapter 3) and have analyzed certain functional identities acting on -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 -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 (considered as a subring of its maximal left quotient ring ). 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 -freeness of rings that contain elements satisfying certain technical conditions — specifically, the so-called fractional degree of such elements must be ≥ . In Section 5.3 we shall see that the basic result on -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.

Every associative ring can be turned into a Lie ring by introducing a new product [] = . So we may regard simultaneously as an associative ring and as a Lie ring. What is the connection between the associative and the Lie structure of ? This question has been studied for more than fifty years by numerous authors, most notably by Herstein and many of his students (see, for example, [, , ]). One of the first questions that one might ask in this context is: If rings and are isomorphic as Lie rings, are they then also isomorphic (or at least antiisomorphic) as associative rings? In more technical terms one can rephrase this question as whether a Lie isomorphism : → always “arises” from an (anti)isomorphism. This is just the simplest question that one can ask in this setting. More general (and from the point of view of the theory of Lie algebras also more natural) questions concern the structure of Lie homomorphisms between various Lie subrings of associative rings. Analogous problems can be formulated for Lie derivations.

“Linear Preserver Problems” are a very popular research area especially in Linear Algebra, and also in Operator Theory and Functional Analysis. These problems deal with linear maps between algebras that, roughly speaking, preserve certain properties; the goal is to find the form of these maps. This is indeed a rather vague description, and certainly one could explain what is a linear preserver problem in a more precise and systematic manner. But let us instead give a couple of illustrative examples.

In this closing chapter we consider three rather unrelated applications of FI’s. The common property of all three topics is the Lie algebra framework. But otherwise, each of them has a different background. We shall discuss the motivation and history just briefly (mostly at the end of the chapter), and in each section we will introduce the necessary notions in a very concise manner. It is not our intention to go into the heart of the matter of these topics. Our main goal is to indicate that FI’s are hidden behind various mathematical notions, and after tracing them out one can effectively apply the theory presented in Part II.