Let be a division ring with a -derivation , where is an endomorphism of and be the quotient division ring of the Ore extension over in an indeterminate . First, we describe non-commutative valuation rings of which contain . Suppose that is compatible with , where is a total valuation ring of , then , the localization of at , is a total valuation ring of . Applying the description above, then, second, we describe non-commutative valuation rings of such that and , which is the aim of this paper. In the end of each section we give several examples to display some of the various phenomena.
Guangming XIE. Hidetoshi MARUBAYASHI. Shigeru KOBAYASHI. Hiroaki KOMATSU. "Non-commutative valuation rings of over a division ring $K$." J. Math. Soc. Japan 56 (3) 737 - 752, July, 2004. https://doi.org/10.2969/jmsj/1191334084