Identities involving additive maps on division rings

TL;DR

The paper characterizes additive maps on division rings under functional identities, formulating the problem via generalized polynomial identities in $D_G[Y]$ and proving a dichotomy: a nontrivial identity forces either an elementary-operator structure or finite-dimensionality over the center. It proves a general result for the identity $G_1(y) g(y) G_2(y) = H(y)$ and uses it to deduce that, for a noncommutative division ring $D$ with $ ext{char}(D) eq 2$, additive maps $g_1,g_2$ satisfying $g_1(y) y^{-m} + y^n g_2(y^{-1}) = 0$ must be zero when $(m,n) eq (1,1)$, with precise characteristic-induced exceptions. It then analyzes $g(y^2)$-type identities and extends the framework to solve the main inverse-type equation, showing that, aside from specific positive-characteristic cases with $p-1 ig| n+m-2$, the only additive solutions are trivial. Overall, the work advances the understanding of functional identities for additive maps on division rings, linking GPI techniques with classical ring theory tools such as Jacobson’s theorem and Hua’s identity.

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 $(m,n) \neq (1,1)$.

Theorems & Definitions (62)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 1.4
• Theorem 1.5
• Theorem 1.6
• Corollary 1.7
• proof
• Example 1.8
• Example 1.9