A class of functional equations for additive functions

The study of functional equations in which the unknown functions are assumed to be additive has a long history and continues to be an active area of research. Here we discuss methods for solving functional equations of the form ( \(*\) $\ast$ ) \(\sum _{j=1}^{k} x^{p_j}f_j(x^{q_j}) = 0\) ${\sum }_{j=1}^k{x}^{p_j}{f}_j(x^{q_j})=0$ , where the \(p_j,q_j\) $p_j,q_j$ are non-negative integers, the \(f_j:R \rightarrow S\) $f_j:R\to S$ are additive functions, S is a commutative ring, and R is a sub-ring of S. This area of research has ties to commutative algebra since homomorphisms and derivations satisfy equations of this type. Methods for solving all homogeneous equations of the form ( \(*\) $\ast$ ) can be found in Ebanks (Aequ Math 89(3):685-718, 2015), Ebanks (Results Math 73(3):120, 2018) and Gselmann et al. (Results Math 73(2):27, 2018). It seems that this fact may have been overlooked, judging by some results about a particular case of ( \(*\) $\ast$ ) in recent publications. We also present a new method for the homogeneous case by combining the results above with [6], and we show how to solve non-homogeneous equations of the form ( \(*\) $\ast$ ).