Remarks on the notion of homo-derivations

Abstract. The purpose of this paper is to study the (different) notions of homo-derivations. These are additive mappings \(f\) of a ring \(R\) that also fulfill the identity $$ f(xy)=f(x)y+xf(y)+f(x)f(y) \quad \left(x, y\in R\right), $$ or (in case of the other notion) the system of equations $$ \begin{array}{rcl} f(xy)&=&f(x)f(y)\\ f(xy)&=&f(x)y+xf(y) \end{array} \qquad \left(x, y\in R\right). $$ Our primary aim is to investigate the above equations without additivity as well as the following Pexiderized equation $$ f(xy)=h(x)h(y)+xk(y)+k(x)y. $$ The obtained results show that under rather mild assumptions homo-derivations can be fully characterized, even without the additivity assumption.