Remarks on the notion of homo-derivations

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 $$f(xy)=f(x)y+xf(y)+f(x)f(y) \qquad \left(x, y\in R\right),$$ or (in case of the other notion) the system of equations $$f(xy)=f(x)f(y)$$ $$f(xy)=f(x)y+xf(y) \qquad \left(x, y\in R\right).$$ 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.