Let be a total valuation ring of a division ring with an automorphism and let be a graded extension of in , the skew Laurent polynomial ring. We classify by distinguishing three different types based on the properties of and , and a complete description of for all is given in the case where is not a finitely generated left -ideal.
"A classification of graded extensions in a skew Laurent polynomial ring, II." J. Math. Soc. Japan 61 (4) 1111 - 1130, October, 2009. https://doi.org/10.2969/jmsj/06141111