Some Functional Identities on Rings
Project no. 22955 · Status: Kapanmış · 2022
Summary
<p>A functional identity on a ring (or an algebra) can be defined as an identical relation involving arbitrary mappings of the ring that is satisfied by every element of the ring (or, in general, of an appropriate subset of the ring). The goal of the theory is to describe the form of the maps involved in the identity, or, when this is not possible, to determine the structure of the ring. Early works on functional identities considered those identities involving some special types of mappings (such as derivations, automorphisms, Lie or a Jordan mappings) of prime rings.</p><p>During the last two decades, a lot of work has been done concerning various local properties of additive maps of rings. In this project, we firstly considered a certain functional identity involving additive maps of division and local rings which is satisfied on the identity-product of the variables 𝑥𝑥 and 𝑦𝑦, and subsequently the forms of the mappings involved in the identity were determined. Next, a complete characterization of Lie anti-derivations of a prime ring R were determined under the assumption that 𝑅𝑅 does not satisfy the standard polynomial identity 𝑠_4 of four variables. As a byproduct of this result, we also determined all possible forms of Lie derivable pairs of additive maps of prime rings under some mild assumptions.</p>