Noncommutative Valuation Rings of the Quotient Artinian Ring of a Skew Polynomial Ring

Let R be a Dubrovin valuation ring of a simple Artinian ring Q and let QX,] be the skew polynomial ring over Q in an indeterminate X, where is an automorphism of Q. Consider the natural map from QX,]_XQX,] to Q, where QX,]_XQX,] is the localization of QX,] at the maximal ideal XQX,] and set , the complete inverse image of R by . It is shown that is a Dubrovin valuation ring of Q(X,) (the quotient ring of QX,]) and it is characterized in terms of X and Q. In the case where R is an invariant valuation ring, the given automorphism is classified into five types, in order to study the structure of (the value group of ). It is shown that there is a commutative valuation ring R with automorphism which belongs to each type and which makes Abelian or non-Abelian. Furthermore, some examples are used to show that several ideal-theoretic properties of a Dubrovin valuation ring of Q with finite dimension over its center, do not necessarily hold in the case where Q is infinite-dimensional. Presented by A. VerschorenMathematics Subject Classifications (2000) 16L99, 16S36, 16W60.