Derivation-Simple Algebras and the Structures of Generalized Lie Algebras of Witt Type

We classify all the pairs of a commutative associative algebra with an identity element and its finite-dimensional commutative locally-finite derivation subalgebra such that the commutative associative algebra is derivation-simple with respect to the derivation subalgebra over an algebraically closed field with characteristic 0. Such pairs are the fundamental ingredients for constructing generalized simple Lie algebras of Cartan type. Moreover, we determine the isomorphic classes of the generalized simple Lie algebras of Witt Type. The structure space of these algebras is given explicitly.