  Let V be a total valuation ring of a division ring K, Q be the additive group
of the rational numbers, Aut(K) be the group of automorphisms of K. Let sigma
be a group homomorphism from Q to Aut(K). Let K[Q, sigma ] be the skew group
ring of Q over K. In this paper, we classify graded extensions of V in K[Q,
sigma] into two types and study the structure of them.
