What is a Functional Identity?
Part of the
Frontiers in Mathematics
book series (FM)
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.
KeywordsPrime Ring Division Ring Associative Ring Functional Identity Unital Ring
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