The range and kernel inclusion of algebraic derivations and commuting maps

Let δ and δ′ be derivations of an algebra A. Consider the following conditions: (i) the range of δ′ is contained in the range of δ, (ii) the kernel of δ is contained in the kernel of δ′, (iii) δ is an inner derivation generated by a ∈ A and δ′ is an inner derivation generated by a polynomial in a. If δ is algebraic then we show that under rather mild additional assumptions these three conditions are closely related. This is used in obtaining some new characterizations of commuting maps on certain rings and normed algebras.