On modular extensions

M. E. Sweedler has proved that modular extensions of fields are characterized by a tensor product of primitive elements, and also by the equivalent condition that the ground field is the fixed field under higher derivations. In this paper we shall give an extension of his work about modular field extensions to modular integral domain extensions. Moreover, we shall prove that a modular extension is an extension that the derivation algebra is generated by components of higher derivations under some conditions. For example, in a finite extension of a field, a modular extension is characterized by the fact that the derivation algebra is generated by components of higher derivations.