Certain functional identities on division rings

Abstract

We study the functional identity G(x)f(x)=H(x) on a division ring D, where f:D→D is an additive map and G(X)≠0,H(X) are generalized polynomials in the variable X with coefficients in D. Precisely, it is proved that either D is finite-dimensional over its center or f is an elementary operator. Applying the result and its consequences, we prove that if D is a noncommutative division ring of characteristic not 2, then the only solution of additive maps f,g on D satisfying the identity f(x)=xng(x−1) with n≠2 a positive integer is the trivial case, that is, f=0 and g=0. This extends Catalano and Merchán's result in 2023 to get a complete solution.