Identities involving additive maps on division rings

Lovepreet Singh; S. K. Tiwari

paper

Identities involving additive maps on division rings

arXiv:2412.19223

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 $char (D) \neq = 2$ , then the only possible solution of additive maps $g_{1}, g_{2} : D \to D$ 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 $(m, n) \neq = (1, 1)$ .