A classification of graded extensions in a skew Laurent polynomial ring

Let V be a total valuation ring of a division ring K with an automorphism or and let A = circle plus i is an element of zA(i)X(i) be a graded extension of V in K[X, X-1; sigma], the skew Laurent polynomial ring. We classify A by distinguishing four different types based on the properties of A, and A(-1). A complete description of A(i) for all i is an element of Z is given in the case where A(1) is a finitely generated left O-l(A(1))-ideal.