Projects / Programmes
Mappings of rings and algebras II
| Code |
Science |
Field |
Subfield |
| 1.01.01 |
Natural sciences and mathematics |
Mathematics |
Analysis |
| Code |
Science |
Field |
| P120 |
Natural sciences and mathematics |
Number theory, field theory, algebraic geometry, algebra, group theory |
| P140 |
Natural sciences and mathematics |
Series, Fourier analysis, functional analysis |
functional identity, ring, algebra, homomorphism, derivation, Banach algebra, operator
Researchers (6)
Organisations (1)
Abstract
The main goal of this research is to develop the theory of functional identities in rings and to find its applications to different topics. A functional identity is, roughly speaking, an identity, holding for all elements in a ring (or some appropriate subset of a ring), which involves arbitrary mappings. When considering a functional identity, the goal is to find either the form of all maps involved or to get some information on the structure of the ring. Similarly we consider the so-called generalized functional identities where, besides arbitrary mappings, some fixed elements of the ring also appear. The concept of a functional identity can be viewed as a generalization of the concept of a polynomial identity, while the concept of a generalized functional identity extends the concept of a generalized polynomial identity. In view of the applications, however, we see that (generalized) functional identities can be considered more as a complement to the theory of (generalized) polynomial identities. Namely, definite results on functional identities can be obtained especially in those rings that do not satisfy polynomial identities. There are various problems where functional identities have been proved more efficient than other methods known so far. In particular, this holds for long-standing Herstein''s conjectures on Lie isomorphisms. One of the main purposes of the present research is to give the final settlement of these conjectures. Algebraic methods and results, obtained in our research, are also applicable to some other mathematical areas, in particular, to functional analysis and operator theory.
