Identities involving additive maps on division rings

TL;DR

This paper investigates functional identities involving additive maps on division rings, establishing conditions under which such maps must be trivial, especially in non-commutative division rings with characteristic not equal to two.

Contribution

It characterizes solutions to specific additive map identities on division rings, extending understanding of their structure and proving triviality under certain conditions.

Findings

Additive maps satisfying the identity are trivial in non-commutative division rings with char ≠ 2.

The paper generalizes functional identities involving generalized polynomials.

It provides conditions for the triviality of solutions to specific additive map equations.

Abstract

Let $g$ be an additive map on a division ring $D$. In this paper, we study the functional identity $G_{1}(y)g(y)G_{2}(y) = H(y)$, where $G_{1}(Y), G_{2}(Y)$, $H(Y)$ are generalized polynomials in $D_{G}[Y]$ such that both $G_{1}(Y)$ and $G_{2}(Y)$ are non-zero. By application of this result and its implications, we prove that if $D$ is a non-commutative division ring with $\operatorname{char}(D) \neq 2$, then the only possible solution of additive maps $g_{1},g_{2}: D \rightarrow D$ satisfying the identity $g_{1}(y)y^{-m} + y^{n}g_{2}(y^{-1})= 0$ is $ g_{1} = g_{2} = 0$, where $m$ and $n$ are positive integers with $(m,n) \neq (1,1)$.