Fueter and Moisil’s definition of right regular functions in an algebra $A$ depends on the choice of a basis in $A$. We determine here the group $G$ of linear transformations (changes of basis in $A$) which map the set of right regular functions into itself. $G$ is strictly related to the reducibility of $A$ as a direct sum of left ideals, and contains the subgroup $G_{0}$ of all similarity transformations with $0$ as a fixed-point. In particular, we show that $G_{0} = G$ in the case of quaternion and Clifford algebras.